Home Chain conditions on composite Hurwitz series rings
Article Open Access

Chain conditions on composite Hurwitz series rings

  • Jung Wook Lim and Dong Yeol Oh EMAIL logo
Published/Copyright: September 22, 2017
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

In this paper, we study chain conditions on composite Hurwitz series rings and composite Hurwitz polynomial rings. More precisely, we characterize when composite Hurwitz series rings and composite Hurwitz polynomial rings are Noetherian, S-Noetherian or satisfy the ascending chain condition on principal ideals.

Keywords: Composite Hurwitz series ring H(R, D); Composite Hurwitz polynomial ring h(R, D); Noetherian ring; S-Noetherian ring; Présimplifiable ring; Ascending chain condition on principal ideals
MSC 2010: 13A15; 13E05; 13E99; 13F15

1 Introduction

1.1 Hurwitz series rings

The formal power series rings and polynomial rings have been of interest and have had important applications in many areas, one of which has been differential algebra. In [1], Keigher introduced a variant of the ring of formal power series and studied some of its properties. In [2], Keigher called such a ring the ring of Hurwitz series and examined its ring theoretic properties. Since then, many works on the ring of Hurwitz series have been done ([3–5]).

Let R be a commutative ring with identity, R〚X〛 (resp., R[X]) the formal power series ring (resp., polynomial ring) over R, and H(R) the set of formal expressions of the form ∑n=0∞anXn, where an ∈ R. Define addition and *-product on H(R) as follows: for f=∑n=0∞anXn,g=∑n=0∞bnXn∈H(R),

f+g=∑n=0∞(an+bn)Xn and f∗g=∑n=0∞cnXn,

where cn=∑k=0n(nk)akbn−k and (nk)=n!(n−k)!k! for nonnegative integers n ≥ k. Then H(R) becomes a commutative ring with identity containing R under these two operations, i.e., H(R) = (R〚X〛, +, *). The ring H(R) is called the Hurwitz series ring over R. The Hurwitz polynomial ring h(R) over R is the subring of H(R) consisting of formal expressions of the form ∑k=0nakXk, i. e., h(R) = (R[X], +, *).

Let R ⊆ D be an extension of commutative rings with identity, and let H(R, D) = {f ∈ H(D)| the constant term of f belongs to R} (resp., h(R, D) = {f ∈ h(D)| the constant term of f belongs to R}). Then H(R, D) (resp., h(R, D)) is a commutative ring with identity. We call H(R, D) (resp., h(R, D)) a composite Hurwitz series ring (resp., composite Hurwitz polynomial ring). More precisely, H(R, D) (resp., h(R, D)) is a subring of H(D) (resp., h(D)) containing H(R) (resp., h(R)), i. e., H(R, D) = (R + XD〚X〛, +, *) (resp., h(R, D) = (R + XD[X], +, *)) where R + XD〚X〛 = {f ∈ D〚X〛| the constant term of f belongs to R} (resp., R + XD[X] = {f ∈ D[X]| the constant term of f belongs to R}). Hence if R ⊊ D, then H(R, D) (resp., h(R, D)) gives algebraic properties of Hurwitz series (resp., Hurwitz polynomial) type rings strictly between two Hurwitz series rings (resp., Hurwitz polynomial rings). Also, it is easy to see that H(R, D) (resp., h(R, D)) is a pullback of R and H(D) (resp., h(D)).

1.2 Noetherian rings and related rings

Chain conditions have for many years been important tools in commutative algebra and algebraic geometry because of their use in producing many theorems and applications. For example, a relation between the ascending chain conditions on ideals and finitely generatedness of ideals in rings permits an interesting measure of the size and behavior of such rings, and the Noetherian condition plays a significant role to prove many results on varieties, homology and cohomology. Recently, Anderson and Dumitrescu [6] introduced the notion of S-Noetherian rings and gave a number of S-variants of well-known results for Noetherian rings. After them, S-Noetherian rings have been studied by some mathematicians (see [7–11]).

In [10, 12–14], the authors characterized when composite rings R + XD〚X〛 and R + XD[X] are Noetherian rings, S-Noetherian rings, or satisfy the ascending chain condition on principal ideals. It was shown that R + XD〚X〛 (resp., R + XD[X]) is a Noetherian ring if and only if R is a Noetherian ring and D is a finitely generated R-module [13, Theorem 4] (or [12, Proposition 2.1]) (resp., [12, Proposition 2.1] (or [14, Corollary 2.2])); and that R + XD[X] (resp., R + XD〚X〛) is an S-Noetherian ring if and only if R is an S-Noetherian ring and D is an S-finite R-module [10, Theorems 3.6 and 4.4]. Also, it was shown that if D is a présimplifiable ring, then R + XD〚X〛 satisfies the ascending chain condition on principal ideals if and only if U(D) ∩ R = U(R) and for each sequence (dn)n≥ 1 of D with the property that for each n ≥ 1, dn = dn+1rn for some rn ∈ R, d1D ⊆ d2 D ⊆ ⋯ is stationary [12, Proposition 4.21].

In [4], Benhissi and Koja studied when the Hurwitz rings H(R) and h(R) are Noetherian rings, S-Noetherian rings or satisfy the ascending chain condition on principal ideals. They showed that if char (R) = 0, then H(R) is a Noetherian ring if and only if h(R) is a Noetherian ring, if and only if R a Noetherian ring containing ℚ [4, Corollary 7.7]. Also, they proved that for an anti-Archimedean subset S of R with zero characteristic containing an element s0 ∈ S divisible in R by all the nonzero positive integers, if R is an S-Noetherian ring, then h(R) is an S-Noetherian ring [4, Theorem 9.4]; and if R is an S-Noetherian ring and S consists of nonzerodivisors, then H(R) is an S-Noetherian ring [4, Theorem 9.6].

In this paper, we study chain conditions on composite Hurwitz series rings H(R, D) and composite Hurwitz polynomial rings h(R, D), where R ⊆ D is an extension of commutative rings with identity. In Section 2, we give necessary and sufficient conditions for the rings H(R, D) and h(R, D) to be Noetherian rings. We show that if char (R) = 0, then H(R, D) is a Noetherian ring if and only if h(R, D) is a Noetherian ring, if and only R is a Noetherian ring and D is a finitely generated R-module containing ℚ. In Section 3, we give equivalent conditions for the rings H(R, D) and h(R, D) to be S-Noetherian rings, where S is an anti-Archimedean subset of R. We show that if char (R) = 0 and S is an anti-Archimedean subset of R consisting of nonzerodivisors of D which contains an element divisible in D by all the positive integers, then H(R, D) is an S-Noetherian ring if and only if R is an S-Noetherian ring and D is an S-finite R-module; and if char (R) = 0 and S is an anti-Archimedean subset of R which contains an element divisible in D by all the positive integers, then h(R, D) is an S-Noetherian ring if and only if R is an S-Noetherian ring and D is an S-finite R-module. In Section 4, we study when the rings H(R, D) and h(R, D) are présimplifiable. We prove that H(R, D) is présimplifiable if and only if Z(D) ∩ R ⊆ 1 + U(R), where Z(D) is the set of zero-divisors of D and U(R) is the set of units in R. We also prove that if D is a torsion-free ℤ-module, then h(R, D) is présimplifiable if and only if D is a domainlike ring. Finally, in Section 5, we characterize when the rings H(R, D) and h(R, D) satisfy the ascending chain condition on principal ideals. We show that if D is a présimplifiable ring, then H(R, D) satisfies the ascending chain condition on principal ideals if and only if U(D) ∩ R = U(R) and for each sequence (dn)n≥1 of D with the property that for each n ≥ 1, there exists an element rn ∈ R such that dn = dn+1rn, d1 D ⊆ d2 D ⊆ ⋯ is stationary; and if D is a présimplifiable ring with char (D) > 0, then h(R, D) satisfies the ascending chain condition on principal ideals if and only if U(D) ∩ R = U(R) and for each sequence (dn)n≥1 of D with the property that for each n ≥ 1, there exists an element rn ∈ R such that dn = dn+1rn, d1 D ⊆ d2 D ⊆ ⋯ is stationary.

2 Noetherian rings

Let R be a commutative ring with identity. Then the mapping ψ : R〚X〛 → H(R) (resp., ϕ : R[X] → h(R)) defined by

ψ(∑n=0∞anXn)=∑n=0∞n!anXn(resp.,ϕ(∑k=0nakXk)=∑k=0nk!akXk)

is a ring homomorphism [2, Proposition 2.3]; and ψ is an isomorphism if and only if ϕ is an isomorphism, if and only if the ℤ-module R is divisible and torsion-free, if and only if R contains ℚ, where ℚ is the field of rational numbers ([2, Proposition 2.4] and [4, Theorem 1.4 and Corollary 1.5]).

We start this section with the following simple observation without proof.

Lemma 2.1

Let R ⊆ D be an extension of commutative rings with identity. Then the following conditions are equivalent.

  1. D contains ℚ.

  2. The ℤ-module D is divisible and torsion-free.

  3. The mapping ψ : R + XD〚X〛 → H(R, D) defined by ψ (∑n=0∞anXn)=∑n=0∞n!anXn is a ring isomorphism.

  4. The mapping ϕ : R + XD[X] → h(R, D) defined by (∑k=0nakXk)=∑k=0nk!akXk is a ring isomorphism.

Let R ⊆ D be an extension of commutative rings with identity, and set XD〚X〛 = {f ∈ H(R, D)| the constant term of f is zero} (resp., XD[X] = {f ∈ h(R, D)| the constant term of f is zero Then it is easy to see that XD〚X〛 (resp., XD[X]) is an H(R, D)-module (resp., h(R, D)-module).

We are now ready to study when composite Hurwitz rings H(R, D) and h(R, D) are Noetherian rings.

Theorem 2.2

Let R ⊆ D be an extension of commutative rings with identity. If char (R) = 0, then the following statements are equivalent.

  1. H(R, D) (resp., h(R, D)) is a Noetherian ring.

  2. R is a Noetherian ring and D is a finitely generated R-module containing ℚ.

  3. R is a Noetherian ring and XD〚X〛 (resp., XD[X]) is a Noetherian H(R, D)-module (resp., h(R, D)-module).

Proof

(1) ⇒(2) Suppose that H(R, D) (resp., h(R, D)) is a Noetherian ring, and let p be any prime number. Since (X, X2, …) is finitely generated, there exists a positive integer n such that Xpn ∈ (X, X2, …, Xpn−1); so we can find suitable elements g1, …, gpn−1 ∈ H(R, D) (resp., g1, …, gpn−1 ∈ h(R, D)) such that Xpn = X * g1 + ⋯ + Xpn−1 * gpn−1. Comparing the coefficients of Xpn in both sides, we get

1=pn1b1+⋯+pnpn−1bpn−1

for some b1, …, bpn−1 ∈ D. Note that p divides (pnk) for all k = 1, …, pn − 1[4, Lemma 7.3]; so p is a unit in D. Since all the prime numbers are units in D, all the nonzero integers are also units in D. Therefore D contains ℚ, and hence by Lemma 2.1, R + XD〚X〛 (resp., R + XD[X]) is a Noetherian ring. Thus R is a Noetherian ring and D is a finitely generated R-module [13, Theorem 4] (or [12, Proposition 2.1]) (resp., [12, Proposition 2.1] (or [14, Corollary 2.2])).

(2) ⇒ (1) Assume that R is a Noetherian ring and D is a finitely generated R-module. Then R + XD〚X〛 (resp., R + XD[X]) is Noetherian [13, Theorem 4] (or [12, Proposition 2.1]) (resp., [12, Proposition 2.1] (or [14, Corollary 2.2])). Since D contains ℚ, Lemma 2.1 forces H(R, D) (resp., h(R, D)) to be a Noetherian ring.

(1) ⇔ (3) We first show the composite Hurwitz series ring case. Let u : R → D be the natural injection and v : H(D) → D the canonical projection. Consider the following commutative diagram

H(R,D)=R×DH(D)→R↓u↓H(D)→vD≃H(D)/XD[[X]].

Then H(R, D) is the pullback of u and v. Thus the equivalence follows from [15, Proposition 4.10].

The proof for the composite Hurwitz polynomial ring case is the same as that for the composite Hurwitz series ring case. □ 

When R = D in Theorem 2.2, we obtain

Corollary 2.3

([4, Corollary 7.7]). Let R be a commutative ring with identity. If char (R) = 0, then the following assertions are equivalent.

  1. R is a Noetherian ring containing ℚ.

  2. H(R) is a Noetherian ring.

  3. h(R) is a Noetherian ring.

We next show that in Theorem 2.2, the condition that char (R) = 0 is essential.

Theorem 2.4

Let R ⊆ D be an extension of commutative rings with identity and let E be either H(R, D) or h(R, D). If char (R) > 0, then E is never a Noetherian ring.

Proof

Suppose on the contrary that E is a Noetherian ring. Then (X, X2, …) is a finitely generated ideal of E; so there exists a positive integer q such that (X, X2, …) = (X, X2, …, Xq). Let char (R)=p1k1⋯pmkm, where p1, …, pm are distinct prime numbers. Then we can take a positive integer n such that pin > q for all i = 1, …, m; so Xpin∈(X,X2,…,Xpin−1) for all i = 1, …, m. Therefore for each i = 1, …, m, there exist suitable elements gi1, …, gi(pin−1) ∈ E such that Xpin=X∗gi1+⋯+Xpin−1∗gi(pin−1). By comparing the coefficients of Xpin in both sides, we get

1=pin1bi1+⋯+pinpin−1bi(pin−1)

for some bi1, …, bi(pin−1) ∈ D. Hence we obtain

1=∏i=1m((pin1)bi1+⋯+(pinpin−1)bi(pin−1))ki.

Note that pi divides (pinj) for all i = 1, …, m and all j = 1, …, pin − 1[4, Lemma 7.3]; so char (R) divides 1 in D. Hence 1 = 0 in D, which is a contradiction. Thus E is not a Noetherian ring.□ 

3 S-Noetherian rings

Let R be a commutative ring with identity, S a (not necessarily saturated) multiplicative subset of R, and M a unitary R-module. Recall from [6, Definition 1] that an ideal I of R is S-finite if there exist an element s ∈ S and a finitely generated ideal J of R such that sI ⊆ J ⊆ I; and R is an S-Noetherian ring if each ideal of R is S-finite. Also, we say that the R-module M is S-finite if sM ⊆ F ⊆ M for some s ∈ S and some finitely generated R-module F; and M is S-Noetherian if each R-submodule of M is S-finite.

Our first result in this section gives a necessary condition for composite Hurwitz rings H(R, D) and h(R, D) to be S-Noetherian rings, where R ⊆ D is an extension of commutative rings with identity and S is a multiplicative subset of R.

Proposition 3.1

Let R ⊆ D be an extension of commutative rings with identity, S be a (not necessarily saturated) multiplicative subset of R, and E be either H(R, D) or h(R, D) . If E is an S-Noetherian ring, then the following assertions hold.

  1. S contains an element s divisible in D by all the prime numbers.

  2. char (R) = 0.

Proof

(1) Suppose that E is an S-Noetherian ring. Then (X, X2, …) is S-finite; so there exist s ∈ S and f1, …, fm ∈ (X, X2, …) such that s * (X, X2, …) ⊆ (f1, …, fm). Let p be any prime number. Since f1, …, fm ∈ (X, X2, …), we can find a positive integer n such that s * (X, X2, …) ⊆ (X, X2, …, Xpn−1). Therefore sXpn ∈ (X, X2, …, Xpn−1), and hence we can write sXpn = X * g1 + ⋯ + Xpn−1 * gpn−1 for some g1, …, gpn−1 ∈ E. Comparing the coefficients of Xpn in both sides, we obtain

s=(pn1)d1+⋯+(pnpn−1)dpn−1

for some d1, …, dpn−1 ∈ D. Note that p divides (pnk) for all k = 1, …, pn − 1[4, Lemma 7.3]; so p divides s in D. Thus s is divisible in D by all the prime numbers.

(2) Suppose on the contrary that char (R) ≠ 0, and let char

(R)=p1α1⋯pmαm,

where p1, …, pm are distinct prime numbers. Fix an i ∈ {1, …, m}. Since E is an S-Noetherian ring, by (1), there exist elements Sj ∈ S and dj ∈ D such that Sj = pidj. Since char (R) = char(D), we get

s1α1⋯smαm= char(R)d1α1⋯dmαm=0,

which indicates that 0 ∈ S. However this is absurd. Thus char (R) = 0.□ 

Let R be a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. We say that S is anti-Archimedean if (⋂n≥1 snR) ∩ S ≠ ∅ for every s ∈ S. We also say that an integral domain R is an anti-Archimedean domain if ⋂n≥1 anR ≠ 0 for each 0 ≠ a ∈ R (see [16]). Thus R is an anti-Archimedean domain if and only if R\{0} is an anti-Archimedean subset of R. Clearly, every multiplicative subset consisting of units is anti-Archimedean. Also, if V is a valuation domain with no height-one prime ideal (or equivalently, every nonzero prime ideal of V has infinite height), then V\{0} is an anti-Archimedean subset of V [16, Proposition 2.1]. We next characterize when a composite Hurwitz series ring H(R, D) is an S-Noetherian ring under the assumption that S is an anti-Archimedean subset of R.

Theorem 3.2

Let R ⊆ D be an extension of commutative rings with identity and char (R) = 0, and let S be an anti-Archimedean subset of R consisting of nonzerodivisors of D. If S contains an element divisible in D by all the positive integers, then the following statements are equivalent.

  1. H(R, D) is an S-Noetherian ring.

  2. R is an S-Noetherian ring and XD〚X〛 is an S-Noetherian H(R, D)-module.

  3. R is an S-Noetherian ring and D is an S-finite R-module.

Proof

(1) ⇔ (2) Consider the following commutative diagram

H(R,D)=R×DH(D)→R↓u↓H(D)→vD≃H(D)/XD[[X]],

where u is the natural injection and v is the canonical projection. Then H(R, D) is the pullback of u and v. Thus the equivalence follows from [9, Proposition 2.3].

(1) ⇒ (3) Suppose that H(R, D) is an S-Noetherian ring. Since R is a homomorphic image of H(R, D), R is an S-Noetherian ring [11, Lemma 2.2]. Note that XD〚X〛 is an ideal of H(R, D); so there exist s ∈ S and f1, …, fn ∈ D〚X〛 such that s * XD〚X〛 ⊆ (Xf1, …, Xfn). Therefore for any d ∈ D, we have

s∗dX=Xf1∗g1+⋯+Xfn∗gn

for some g1, …, gn ∈ H(R, D). Comparing the coefficients of X in both sides, we get

sd=f1(0)g1(0)+⋯+fn(0)gn(0),

where for each i = 1, …, n, fi(0) and gi(0) denote the constant terms of fi and gi, respectively. Note that fi(0) ∈ D and gi(0) ∈ R for all i = 1, …, n. Hence sD ⊆ f1(0)R + ⋯ + fn(0)R. Thus D is an S-finite R-module.

(3) ⇒ (1) Suppose that R is an S-Noetherian ring and D is an S-finite R-module. Then D is an S-Noetherian R-module [10, Lemma 3.5(2)]; so D is an S-Noetherian ring. Since D is an S-finite R-module, there exist s ∈ S and d1, …, dm ∈ D such that sD ⊆ d1R + ⋯ + dmR; so we have

s∗H(D)⊆d1∗H(R)+⋯+dm∗H(R)⊆d1∗H(R,D)+⋯+dm∗H(R,D).

Hence H(D) is an S-finite H(R, D)-module. Clearly, S is an anti-Archimedean subset of D. Note that char (D)= char(R) = 0; so by our assumption, H(D) is an S-Noetherian ring [4, Theorem 9.6]. Since H(D) is an S-finite H(R, D)-module, H(R, D) is an S-Noetherian ring [6, Corollary 7] (or [10, Lemma 3.5(3)]).□ 

Recall that if R is an S-Noetherian ring with char (R) = 0 and S is an anti-Archimedean subset of R which contains an element divisible in R by all the positive integers, then h(R) is also an S-Noetherian ring [4, Theorem 9.4]. By combining this result with a similar argument as in the proof of Theorem 3.2, we obtain equivalent conditions for a composite Hurwitz polynomial ring h(R, D) to be an S-Noetherian ring.

Theorem 3.3

Let R ⊆ D be an extension of commutative rings with identity and char (R) = 0, and let S be an anti-Archimedean subset of R. If S contains an element divisible in D by all the positive integers, then the following statements are equivalent.

  1. h(R, D) is an S-Noetherian ring.

  2. R is an S-Noetherian ring and XD[X] is an S-Noetherian h(R, D)-module.

  3. R is an S-Noetherian ring and D is an S-finite R-module.

Remark 3.4

Let R ⊆ D be an extension of commutative rings with identity and S a multiplicative subset of R. If R contains ℚ, then all the integers are units in D; so every element in S is divisible in D by all the positive integers. Hence if R contains ℚ, then it follows from Lemma 2.1 that Theorems 3.2 and 3.3 are nothing but parts of [10, Theorems 4.4 and 3,6].

By Proposition 3.1 and Theorems 3.2 and 3.3, we may ask the following question.

Question 3.5

Do Theorems 3.2 and 3.3 still hold if S contains an element divisible in D by all the prime numbers?

We end this section with an example satisfying the conditions in Theorems 3.2 and 3.3. More precisely, we construct an integral domain R, not containing ℚ, with char (R) = 0 such that there exists an anti-Archimedean subset S of R containing an element divisible in R by all the positive integers.

Example 3.6

Let ℤ be the ring of integers and G the weak direct sum of {Zi}i=1∞ which has the reverse lexicographic order, where Zi = ℤ for all positive integers i. Let {Xi}i=1∞∪{Yi}i=1∞ be a set of indeterminates over ℚ and v be the valuation on Q({Xi}i=1∞,{Yi}i=1∞) induced by the mapping Xi ↦ 0 and Yi ↦ ei, of {Xi}i=1∞∪{Yi}i=1∞ into G, where ei is an element of G whose i-th component is 1 and j-th component is 0 for j ≠ i.

  1. Let V be the valuation ring of v and set V* = V\{0}. Then V is a valuation domain containing ℚ with no height-one prime ideals [17, page 254, Exercise 20] and V* is an anti-Archimedean subset of V[16, Proposition 2.1]. Clearly, V is a V*-Noetherian ring; so V〚X〛 (resp., V[X]) is a V*-Noetherian ring [6, Proposition 10] (resp., [6, Proposition 9]). Since V contains ℚ, H(V) (resp., h(V)) is isomorphic to V〚X〛 (resp., V[X]) [4, Theorem 1.4] (resp., [4, Corollary 1.5]). Hence H(V) (resp., h(V)) is a V*-Noetherian ring.

  2. Let R = ℤ + M, where M is the maximal ideal of V. Then R is an anti-Archimedean ring [16, Proposition 2.6] and each ideal of R is comparable with M under the set theoretic inclusion [17, page 202, Exercise 12]. For any integer n, nR = n ℤ + nM = n ℤ + M contains M (since ℚM = M, nM = M). Hence every element in M is divisible in R by all the positive integers. Note that H(R) (resp., h(R)) is not isomorphic to R〚X〛 (resp., R[X]) because R does not contain ℚ. Since R* = R\{0} is anti-Archimedean subset of R and R is R*-Noetherian, it follows from [4, Theorem 9.6] (resp., [4, Theorem 9.4]) that H(R) (resp., h(R)) is an R*-Noetherian ring.

  3. Let R ⊆ D be an extension of integral domains, where R is defined in (2). If D is a finitely generated R-module, then D is an R*-finite R-module. Hence H(R, D) (resp., h(R, D)) is an R*-Noetherian ring.

4 Présimplifiable rings

Let R be a commutative ring with identity, U(R) the set of units of R, and Z(R) the set of zero-divisors of R. Recall that R is preésimplifiable if whenever a, b ∈ R satisfy ab = a, either a = 0 or b ∈ U(R). It was shown in [18] that R is présimplifiable if and only if Z(R) ⊆ 1 + U(R). In [4], the authors studied when Hurwitz rings H(R) and h(R) are présimplifiable. In this section, we modify some properties of elements (units and nilpotent) of H(R) and h(R) in [4] to give equivalent conditions for composite Hurwitz series rings and composite Hurwitz polynomial rings to be présimplifiable.

Our first result in this section is a necessary and sufficient condition for a composite Hurwitz series ring H(R, D) to be présimplifiable, where R ⊆ D is an extension of commutative rings with identity.

Proposition 4.1

Let R ⊆ D be an extension of commutative rings with identity. Then H(R, D) is présimplifiable if and only if Z(D) ∩ R ⊆ 1 + U(R).

Proof

(⇒) Let r ∈ Z(D) ∩ R. Since H(R, D) is présimplifiable, we obtain

r∈Z(H(R,D))⊆1+U(H(R,D)).

Thus r ∈ 1 + U(R) [19, Lemma 2.2(1)].

(⇐) Let f = ∑i=0∞aiXi ∈ Z(H(R, D Then by the assumption, we obtain

a0∈Z(R)⊆Z(D)∩R⊆1+U(R).

Therefore f − 1 ∈ U(H(R, D)) [19, Lemma 2.2(1)], and hence f ∈ 1 + U(H(R, D)). Thus H(R, D) is présimplifiable.□ 

Remark 4.2

For an extension R ⊆ D of commutative rings with identity, Z(R) ⊆ Z(D) ∩ R. Hence if H(R, D) is présimplifiable, then so is H(R). However, the converse does not hold in general. Consider R = ℤ ⊆ D = ℤ[Y]/(3Y). Then clearly H(R) is présimplifiable but H(R, D) is notprésimplifiable because 3 ∈ Z(H(R, D)) and 3 ∉ 1 + U(H(R, D)).

We next study when a composite Hurwitz polynomial ring h(R, D) is présimplifiable, where R ⊆ D is an extension of commutative rings with identity. To do this, we need two lemmas.

Lemma 4.3

Let R ⊆ D be an extension of commutative rings with identity and f = ∑i=0naiXi ∈ h(R, D). Then the following assertions hold.

  1. f is nilpotent if and only if a0 is nilpotent and for each i = 1, …, n, ai is nilpotent or some power of ai is with torsion.

  2. f is a unit if and only if a0 is a unit in R and for each i = 1, …, n, ai is nilpotent or some power of ai is with torsion.

Proof

  1. The proof is identical to that of [4, Theorem 2.4].

  2. (⇒) Assume that f is a unit in h(R, D). Then we can find an element g = ∑i=0maiXi ∈ h(R, D) such that f * g = 1; so a0b0 = 1. Hence a0 is a unit in R. Since h(R, D) ⊆ h(D), f is a unit in h(D); so for each i = 1, …, n, ai is nilpotent or some power of ai is with torsion [4, Theorem 3.1].

(⇐) Assume that for each i = 1, …, n, ai is nilpotent or some power of ai is with torsion. Then by (1), ∑i=0naiXi is nilpotent in h(R, D). Since a0 is a unit in h(R, D), f is a unit in h(R, D).□ 

Lemma 4.4

Let R ⊆ D be an extension of commutative rings with identity. Then the following assertions are equivalent.

  1. For each f ∈ Z(h(R, D)), there exists an element d ∈ D\{0} such that d * f = 0.

  2. D is a torsion-free ℤ-module.

Proof

(1) ⇒ (2) Let a be any nonzero element in D. If there exists a positive integer n such that na = 0, then 0 = naXn = aX * Xn−1; so Xn−1 is a zero-divisor of h(R, D). By the assumption, we can find an element d ∈ D\{0} such that dXn−1 = 0, which is absurd. Thus D is a torsion-free ℤ-module.

(2) ⇒ (1) Since h(R, D) ⊆ h(D), the implication follows directly from [4, Theorem 4.1].□ 

Let R be a commutative ring with identity. Recall that R is a domainlike ring if every zero-divisor of R is nilpotent. It is easy to see that R is domainlike if and only if (0) is primary.

Proposition 4.5

Let R ⊆ D be an extension of commutative rings with identity. If D is a torsion-free ℤ-module, then h(R, D) is présimplifiable if and only if D is a domainlike ring.

Proof

(⇒) Let d ∈ Z(D). Since h(R, D) is présimplifiable, we have

dX∈Z(h(R,D))⊆1+U(h(R,D));

so −1 + dX ∈ U(h(R, D)). Since D is a torsion-free ℤ-module, d is nilpotent by Lemma 4.3(2). Thus D is a domainlike ring.

(⇐) Let f = ∑i=0naiXi ∈ Z(h(R, D)). Since D is a torsion-free ℤ-module, by Lemma 4.4, there exists an element d ∈ D\{0} such that d * f = 0. Hence for all i = 0, …, n, ai is a zero-divisor of D. By the assumption, ai is nilpotent for all i = 0, …, n; so by Lemma 4.3(1), f is nilpotent. Therefore f − 1 is a unit in h(R, D), and hence f ∈ 1 + U(h(R, D)). Thus h(R, D) is présimplifiable.□ 

We next show that in Proposition 4.5, the condition that D is a torsion-free ℤ-module is essential.

Proposition 4.6

Let R ⊆ D be an extension of commutative rings with identity and char (D) = 0. If D is not a torsion-free ℤ-module, then h(R, D) is never présimplifiable.

Proof

Assume that D is not a torsion-free ℤ-module. Then we can find an element d ∈ D\{0} and a positive integer n such that nd = 0; so X * dXn−1 = ndXn = 0. Hence X ∈ Z(h(R, D)). If h(R, D) is présimplifiable, then Z(h(R, D)) ⊆ 1 + U(h(R, D)); so − 1 + X is a unit in h(R, D). Hence by Lemma 4.3(2), 1 is with torsion. However, this is impossible because char (D) = 0. Thus h(R, D) is not présimplifiable.□ 

Let R ⊆ D be an extension of commutative rings with identity. Then it follows directly from Lemma 4.3(2) that if char (R) > 0, then ∑i=0naiXi ∈ h(R, D) is a unit if and only if a0 is a unit in R. Hence a similar argument as in the proof of Proposition 4.1 shows the following result.

Proposition 4.7

Let R ⊆ D be an extension of commutative rings with identity. If char (D) > 0, then h(R, D) is présimplifiable if and only if Z(D) ∩ R ⊆ 1 + U(R).

5 Rings satisfying ascending chain condition on principal ideals

Let R be a commutative ring with identity. We say that R satisfies the ascending chain condition on principal ideals (ACCP) if there does not exist a strict ascending chain of principal ideals of R. It was shown in [19, Theorem 2.4] that if R ⊆ D is an extension of integral domains with char (D) = 0, then H(R, D) satisfies ACCP if and only if h(R, D) satisfies ACCP, if and only if ⋂n≥1r1 ⋯ rnD = (0) for each infinite sequence (rn)n≥1 consisting of nonzero nonunits of R. In this section, we study an equivalent condition for H(R, D) and h(R, D) to satisfy ACCP, where R ⊆ D is an extension of présimplifiable rings with identity.

Theorem 5.1

Let R ⊆ D be an extension of commutative rings with identity. If D is a présimplifiable ring, then the following statements are equivalent.

  1. H(R, D) satisfies ACCP.

  2. U(D) ∩ R = U(R) and for each sequence (dn)n≥1 of D with the property thatfor each n ≥ 1, there exists an element rn ∈ R such that dn = dn+1rn, d1D ⊆ d2D ⊆ ⋯ is stationary.

Proof

(1) ⇒ (2) Let u ∈ U(D) ∩ R. Then 1uX∗H(R,D)⊆1u2X∗H(R,D) ⊆ ⋯ is an ascending chain of principal ideals of H(R, D). Since H(R, D) satisfies ACCP, there exists a positive integer m such that 1um+1 X* H(R, D) = 1um X * H(R, D). Hence u ∈ U(R). Clearly, U(R) ⊆ U(D) ∩ R, which shows that U(D) ∩ R = U(R).

Let (dn)n≥1 be a sequence of D with the property that for each n ≥ 1, there exists an element rn ∈ R such that dn = dn+1rn. Then d1X * H(R, D) ⊆ d2X * H(R, D) ⊆ ⋯ is an ascending chain of principal ideals of H(R, D). Since H(R, D) satisfies ACCP, the chain d1X * H(R, D) ⊆ d2 X * H(R, D) ⊆ ⋯ is stationary; so we can find a positive integer m such that dnX * H(R, D) = dmX * H(R, D) for all n ≥ m. Hence dnD = dmD for all n ≥ m. Thus the chain d1D ⊆ d2D ⊆ ⋯ stops.

(2) ⇒ (1) Let f1 * H(R, D) ⊆ f2 * H(R, D) ⊆ ⋯ be an ascending chain of nonzero principal ideals of H(R, D). Then for each n ≥ 1, fn = fn+1 * gn for some gn ∈ H(R, D). If fn is a unit for some n ≥ 1, then there is nothing to prove; so we assume that fn is a nonunit for all n ≥ 1. For each n ≥ 1, write fn = ∑m=kn∞anmXm and gn = ∑m=0∞bnmXm , where ankn ≠ 0. Since fn is a multiple of fn+1, k1 ≥ k2 ≥ ⋯ ≥ 0; so there exists a positive integer q such that kn = kq for all n ≥ q. Hence ankn = an+1kn+1bn0 for all n ≥ q. By the assumption, the chain aqkqD ⊆ aq+1kq+1 D ⊆ ⋯ is stationary; so we can find an integer p ≥ q such that amkmD = apkpD for all m ≥ p. Therefore for each n ≥ p, there exists an element dn ∈ D such that an+1kn+1 = ankndn. Hence ankn = ankndnbn0 for all n ≥ p. Since D is présimplifiable and ankn ≠ 0, dnbn0 is a unit in D, which indicates that bn0 ∈ U(D) ∩ R = U(R). Hence gn is a unit in H(R, D) [19, Lemma 2.2(1)], which shows that fn * H(R, D) = fp * H(R, D) for all n ≥ p. Thus H(R, D) satisfies ACCP.□ 

Let R ⊆ D be an extension of commutative rings with identity. Note that by Lemma 4.3(2), if char (R) > 0, then ∑i=0naiXi ∈ h(R, D) is a unit if and only if a0 is a unit in R. Hence a similar argument as in the proof of Theorem 5.1 shows the following result.

Theorem 5.2

Let R ⊆ D be an extension of commutative rings with identity. If D is a présimplifiable ring with char (D) > 0, then the following statements are equivalent.

  1. h(R, D) satisfies ACCP.

  2. U(D) ∩ R = U(R) and for each sequence (dn)n≥1 of D with the property thatfor each n ≥ 1, there exists an element rn ∈ R such that dn = dn+1rn, d1D ⊆ d2 D ⊆ ⋯ is stationary.

When R = D in Theorems 5.1 and 5.2, we obtain

Corollary 5.3

Let R be a présimplifiable ring with identity. Then the following assertions hold.

  1. R satisfies ACCP if and only if H(R) satisfies ACCP.

  2. If char (R) > 0, then R satisfies ACCP if and only if h(R) satisfies ACCP.

We are closing this paper with an example which shows that if a ring has characteristic zero, then ACCP property does not ascend into the Hurwitz polynomial ring extension.

Example 5.4

Let K be a field with char (K) = 0, {An}n=1∞ a set of indeterminates over K, and set S = K[{An}n=1∞]/({An+1(An−An+1)}n=1∞). Let an be the image of An in S and R be the localization of S at the ideal (a1, a2,…)S.

  1. R is a présimplifiable ring which satisfies ACCP ([12, Remark 4.17] and [20, Example]).

  2. Note that for all n ≥ 1, anX + 1 = (an+1X + 1) * ((an − an+1)X + 1); so (a1X + 1) * h(R) ⊆ (a2X + 1) * h(R) ⊆ ⋯ is an ascending chain of principal ideals of h(R). Suppose on the contrary that the chain stops. Then there exists a positive integer m such that am+1X + 1 = (amX + 1) * f for some f ∈ h(R). Now, an easy calculation shows that f = ∑n=0∞bnXn , where b0 = 1 and bn = (−1)n+1 n!amn−1(am+1−am) for all n ≥ 1. Since char (K) = 0, bn ≠ 0 for all nonnegative integers n [20, Example]. Hence f2h(R), which is absurd. Thus h(R) does not satisfy ACCP.

Acknowledgement

The authors would like to thank referees for several valuable comments and suggestions. The corresponding author (D.Y.Oh) was supported by Research Fund from Chosun University, 2014.

References

[1] Keigher W. F., Adjunctions and comonads in differential algebra, Pacific J. Math., 1975, 59, 99-112.10.2140/pjm.1975.59.99Search in Google Scholar

[2] Keigher W.F., On the ring of Hurwitz series, Comm. Algebra, 1997, 25, 1845-1859.10.1080/00927879708825957Search in Google Scholar

[3] Benhissi A., Ideal structure of Hurwitz series ring, Contrib. Alg. Geom., 1997, 48, 251-256.Search in Google Scholar

[4] Benhissi A., Koja F., Basic properties of Hurwitz series rings, Ric. Mat., 2012, 61, 255-273.10.1007/s11587-012-0128-2Search in Google Scholar

[5] Liu Z., Hermite and PS-rings of Hurwitz series, Comm. Algebra, 2000, 28, 299-305.10.1080/00927870008841073Search in Google Scholar

[6] Anderson D.D., Dumitrescu T., S-Noetherian rings, Comm. Algebra, 2002, 30, 4407-4416.10.1081/AGB-120013328Search in Google Scholar

[7] Baeck J., Lee G., Lim J.W., S-Noetherian rings and their extensions, Taiwanese J. Math., 2016, 20, 1231-1250.10.11650/tjm.20.2016.7436Search in Google Scholar

[8] Lim J.W., A note on S-Noetherian domains, Kyungpook Math. J., 2015, 55, 507-514.10.5666/KMJ.2015.55.3.507Search in Google Scholar

[9] Lim J.W., Oh D.Y., S-Noetherian properties on amalgamated algebras along an ideal, J. Pure Appl. Algebra, 2014, 218, 1075-1080.10.1016/j.jpaa.2013.11.003Search in Google Scholar

[10] Lim J.W., Oh D.Y., S-Noetherian properties of composite ring extensions, Comm. Algebra, 2015, 43, 2820-2829.10.1080/00927872.2014.904329Search in Google Scholar

[11] Liu Z., On S-Noetherian rings, Arch. Math. (Brno), 2007, 43, 55-60.Search in Google Scholar

[12] Hizem S., Chain conditions in rings of the form A + XB[X] and A + XI[X], in: M. Fontana, et al. (Eds.), Commutative Algebra and Its Applications: Proceedings of the Fifth International Fez Conference on Commutative Algebra and Its Applications, Fez, Morocco, W. de Gruyter Publisher, Berlin, 2008, pp. 259-274.Search in Google Scholar

[13] Hizem S., Benhissi A., When is A + XB〚X〛 Noetherian?, C. R. Acad. Sci. Paris, 2005, 340, 5-7.10.1016/j.crma.2004.11.017Search in Google Scholar

[14] Lim J.W., Oh D.Y., Chain conditions in special pullbacks, C. R. Acad. Sci. Paris, 2012, 350, 655-659.10.1016/j.crma.2012.07.003Search in Google Scholar

[15] D’Anna M., Finocchiaro C.A., Fontana M., Amalgamated algebras along an ideal, in: M. Fontana et al. (Eds.), Commutative Algebra and Its Applications: Proceedings of the Fifth International Fez Conference on Commutative Algebra and Its Applications, Fez, Morocco, W. de Gruyter Publisher, Berlin, 2008, pp. 155-172.Search in Google Scholar

[16] Anderson D.D., Kang B.G., Park M.H., Anti-Archimedean rings and power series rings, Comm. Algebra, 1998, 26, 3223-3238.10.1080/00927879808826338Search in Google Scholar

[17] Gilmer R., Multiplicative Ideal Theory, Queen’s Papers in Pure Appl. Math., vol. 90, Queen’s University, Kingston, Ontario, 1992.Search in Google Scholar

[18] Bouvier A., Anneaux présimplifiables, Rev. Roumaine Math. Pures Appl., 1974, 19, 713-724.Search in Google Scholar

[19] Lim J.W., Oh D.Y., Composite Hurwitz rings satisfying the ascending chain condition on principal ideals, Kyungpook Math. J., 2016, 56, 1115-1123.10.5666/KMJ.2016.56.4.1115Search in Google Scholar

[20] Heinzer W.J., Lantz D.C., ACCP in polynomial rings: a counterexample, Proc. Amer. Math. Soc., 1994, 121, 975-977.10.1090/S0002-9939-1994-1232140-9Search in Google Scholar
Received: 2017-4-14
Accepted: 2017-7-25
Published Online: 2017-9-22

© 2017 Lim and Oh

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Articles in the same Issue

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
https://doi.org/10.1515/math-2017-0097
Keywords for this article
Composite Hurwitz series ring H(R, D); Composite Hurwitz polynomial ring h(R, D); Noetherian ring; S-Noetherian ring; Présimplifiable ring; Ascending chain condition on principal ideals
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 29.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2017-0097/html
Scroll to top button