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On nonnil-coherent modules and nonnil-Noetherian modules

  • Younes El Haddaoui , Hwankoo Kim EMAIL logo and Najib Mahdou
Published/Copyright: November 24, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this article, we introduce two new classes of modules over a ϕ -ring that generalize the classes of coherent modules and Noetherian modules. We next study the possible transfer of the properties of being nonnil-Noetherian rings, ϕ -coherent rings, and nonnil-coherent rings in the amalgamated algebra along an ideal.

Keywords: nonnil-coherent ring; ϕ-coherent ring; ϕ-submodule; nonnil-coherent module; nonnil-Noetherian ring; nonnil-Noetherian module
MSC 2010: 13A15; 13C05; 13E15; 13F05

1 Introduction

All rings considered in this article are assumed to be commutative with non-zero identity and prime nilradical. We use Nil ( R ) to denote the set of nilpotent elements of R and Z ( R ) , the set of zero-divisors of R . A ring with Nil ( R ) being divided prime (i.e., Nil ( R ) ⊂ x R for all x ∈ R ⧹ Nil ( R ) ) is called a ϕ -ring. El Khalfi et al. [1], and Chhiti et al. [2] studied when the amalgamation algebra along an ideal is a ϕ -ring. Let ℋ be the set of all rings with divided prime nilradical. A ring R is called a strongly ϕ -ring if R ∈ ℋ and Z ( R ) = Nil ( R ) . Let R be a ring and M be an R -module; we define

ϕ - tor ( M ) = { x ∈ M ∣ s x = 0 for some s ∈ R ⧹ Nil ( R ) } .

If ϕ - tor ( M ) = M , then M is called a ϕ -torsion module, and if ϕ - tor ( M ) = 0 , then M is called a ϕ -torsion-free module. It is worth noting that in the language of torsion theory, the class T of all ϕ -torsion modules is a (hereditary) torsion class, whereas T is closed under (submodules,) direct sums, epimorphic images, and extensions. An ideal I of R is said to be nonnil if I ⊈ Nil ( R ) . An R -module M is said to be ϕ -divisible if M = s M for all s ∈ R ⧹ Nil ( R ) .

Among the many recent generalizations of the concept of a coherent ring in the literature, we can find the following: due to Bacem and Ali [3], a ϕ -ring R is called ϕ -coherent if R / Nil ( R ) is a coherent domain [3, Corollary 3.1]. A ϕ -ring R is said to be nonnil-coherent if every finitely generated nonnil ideal is finitely presented, which is equivalent to saying that R is ϕ -coherent and ( 0 : r ) is a finitely generated ideal of R for each r ∈ R ⧹ Nil ( R ) , where ( 0 : r ) = { x ∈ R ∣ r x = 0 } [4, Proposition 1.3]. In [5], an R -module M is said to be coherent if M is a finitely generated R -module and every finitely generated submodule of M is a finitely presented R -module. In [6], an R -module M is said to be Noetherian if every submodule of M is finitely generated. In [7], Badawi introduced and studied a new class of ϕ -rings, which are said to be nonnil-Noetherian. A ϕ -ring R is said to be nonnil-Noetherian if every nonnil ideal of R is finitely generated, which is equivalent to saying that R / Nil ( R ) is a Noetherian domain ([7, Theorem 2.4]). In 2015, Yousefian Darani [8] introduced a new class of modules that is closely related to the class of Noetherian modules. An R -module M with Nil ( M ) ≔ Nil ( R ) M , a divided prime submodule (i.e., Nil ( M ) , is a prime submodule of M and comparable with each submodule of M ) is said to be nonnil-Noetherian if every nonnil submodule N of M (i.e., N ⊈ Nil ( M ) ) is finitely generated. In 2020, Yousefian Darani and Rahmatinia [9] introduced and studied ϕ -Noetherian modules as a new class of Noetherian modules. A module M is said to be ϕ -Noetherian if Nil ( M ) is divided prime and each submodule that properly contains Nil ( M ) is finitely generated.

Let R be a ring and E an R -module. Then R ∝ E , the trivial ring extension of R by E , is the ring whose additive structure is that of the external direct sum R ⊕ E and whose multiplication is defined by ( a , e ) ( b , f ) ≔ ( a b , a f + b e ) for all a , b ∈ R and all e , f ∈ E (this construction is also known by other terminologies and other notations, such as the idealization R ( + ) E ) (see [5,10, 11,12]).

Let A and B be two rings, let J be an ideal of B and let f : A ⟶ B be a ring homomorphism. In this setting, we can consider the following subring of A × B :

A ⋈ f J = { ( a , f ( a ) + j ) ∣ a ∈ A , j ∈ J } ,

called the amalgamation of A with B along J with respect to f (introduced and studied by D’Anna et al. [13,14]). This construction is a generalization of the amalgamated duplication of a ring along an ideal (introduced and studied by D’Anna and Fontana [15] and denoted by A ⋈ I ).

This article consists of five sections including an Introduction. In Section 2, we introduce and study a new class of modules over a ϕ -ring R which are called nonnil-coherent modules. Let M be an R -module and N be a submodule of M . Then, N is said to be a ϕ -submodule of M if M / N is a ϕ -torsion module (see Definition 2.1). Using Definition 2.1, an R -module M is said to be nonnil-coherent if M is finitely generated and each finitely generated ϕ -submodule of M is finitely presented (see Definition 2.4). We give some properties that characterize these modules. In Section 3, we introduce and study another definition of nonnil-Noetherian modules that is different from the definition of [8,9]. An R -module M is said to be nonnil-Noetherian if M is a finitely generated module and every ϕ -submodule of M is finitely generated (see Definition 3.1). Next, we give some properties that characterize these modules. In Section 4, we study the possible transfer of the properties of nonnil-coherent rings and nonnil-Noetherian rings in trivial ring extensions. In the last section, we study the possible transfer of the properties of being ϕ -coherent rings and nonnil-Noetherian rings in an amalgamation algebra along an ideal.

For any undefined terminology and notation, the reader is referred to [5,6,16,17]. Throughout this article, if S is a multiplicative subset of a ring R , then we assume that S ∩ Nil ( R ) = ∅ .

2 On nonnil-coherent modules

In this section, we introduce and study a new class of modules over a ϕ -ring R , which are called nonnil-coherent modules. Recall that in [5], an R -module M is said to be coherent if M is finitely generated and every finitely generated submodule is finitely presented.

Recall that an R -module M is said to be ϕ -torsion if, for all x ∈ M , there exists s ∈ R ⧹ Nil ( R ) such that s x = 0 .

Definition 2.1

Let R ∈ ℋ and M be an R -module. A submodule N of M is said to be a ϕ -submodule if M / N is a ϕ -torsion module.

Example 2.2

A nonnil submodule is not in general a ϕ -submodule. For example, set R ≔ Z , which is a ϕ -ring, and M = C [ X ] as an R -module. It is easy to see that every nonzero subgroup N of M is a nonnil submodule, in particular, the subgroup N = Q [ X ] is a nonnil submodule of M . But for any nonzero s ∈ Z , we obtain s i ∉ N . Hence, N is never a ϕ -submodule of M . Therefore, we deduce that the class of nonnil-submodules of an R -module is different from the class of ϕ -submodules of that R -module.

There is a natural question: If R is a ϕ -ring, then is every submodule N of an R -module M (with Nil ( M ) being prime divided) such that N contains properly Nil ( M ) a ϕ -submodule of M ? The following example shows that the answer to this question is negative.

Example 2.3

Let R = Z , M = C , and N = Q . Then, Nil ( M ) = 0 is a divided prime submodule of M and N properly contains Nil ( M ) , but C / Q is never a torsion abelian group. Therefore, Q is not a ϕ -subgroup of C .

Definition 2.4 allows us to generalize the definition of coherent modules over a ϕ -ring.

Definition 2.4

Let R ∈ ℋ . An R -module M is said to be nonnil-coherent if M is finitely generated and every finitely generated ϕ -submodule of M is finitely presented. In particular, every coherent module over a ϕ -ring is nonnil-coherent.

Remark 2.5

Note that for a ϕ -torsion R -module M , we have

M  is nonnil-coherent ⇔ M  is coherent .

Recall from [18] that an R -module F is said to be ϕ -flat if f ⊗ R F is an R -monomorphism for any R -monomorphism f , where Coker ( f ) is a ϕ -torsion R -module. Recall in [3] that a ϕ -ring is said to be nonnil-coherent if every finitely generated nonnil ideal is finitely presented.

Now, we are able to give a new characterization of nonnil-coherent rings.

Theorem 2.6

The following are equivalent for a ϕ -ring R:

  1. R is a nonnil-coherent ring.

  2. R is a nonnil-coherent R-module.

  3. Every finitely generated free R-module is nonnil-coherent.

  4. Every finitely presented module is nonnil-coherent.

  5. Every finitely generated ϕ -submodule of a finitely presented R-module is finitely presented.

  6. Any direct product of ϕ -flat R-modules is ϕ -flat.

  7. R I is ϕ -flat for any index set I.

Proof

( 6 ) ⇔ ( 7 ) ⇔ ( 1 ) This follows from [3, Theorem 2.4].

( 4 ) ⇒ ( 5 ) Straightforward.

( 5 ) ⇒ ( 1 ) This follows immediately from the fact that every nonnil ideal of R is a ϕ -submodule of R .

( 1 ) ⇒ ( 2 ) Assume that R is a nonnil-coherent ring and let I be a finitely generated ideal of R such that R / I is ϕ -torsion. If I ⊂ Nil ( R ) , then, for any r ∈ R ⧹ Nil ( R ) , there exists s ∈ R ⧹ Nil ( R ) such that s r ∈ I ⊂ Nil ( R ) since R / I is a ϕ -torsion R -module, a desired contradiction since Nil ( R ) is a prime ideal of R . Therefore, I is a nonnil ideal. As R is a nonnil-coherent ring, I is a finitely presented ideal. Therefore, R is a nonnil-coherent R -module.

( 2 ) ⇒ ( 1 ) Let I be a finitely generated nonnil ideal of R . Since R is a nonnil-coherent module and R / I is ϕ -torsion, I is finitely presented. Therefore, R is a nonnil-coherent ring.

( 6 ) ⇒ ( 3 ) Let F be a finitely generated free R -module and N be a finitely generated ϕ -submodule of F . Then, F and F / N are finitely presented R -modules. Since R I is a ϕ -flat module for any index set I , by [18, Theorem 3.2] we obtain the following commutative diagram with exact rows:

Since the two right vertical arrows are isomorphisms by [17, Lemma I.13.2], we obtain N ⊗ R R I ≅ N I , and so N is a finitely presented R -module by [17, Lemma I.13.2]. Therefore, F is a nonnil-coherent R -module.

( 3 ) ⇒ ( 4 ) Let M be a finitely presented R -module. Then, M ≅ F / N , where F is a finitely generated free R -module and N is a finitely generated submodule of F . Let X be a finitely generated ϕ -submodule of M . Then, X ≅ L / N such that L is a finitely generated submodule of F with N ⊂ L . Since F is a nonnil-coherent module and M / X ≅ F / L is ϕ -torsion, L is a finitely presented R -module. Now, it follows immediately from [19, (4.54) Lemma] that X is finitely presented. Therefore, M is a nonnil-coherent module.□

The following theorem characterizes when a finitely generated submodule of a nonnil-coherent module is nonnil-coherent.

Theorem 2.7

Let R ∈ ℋ and M be a nonnil-coherent R -module. If N is a finitely generated ϕ -submodule of M , then N is a nonnil-coherent module.

Before proving Theorem 2.7, we need the following lemma.

Lemma 2.8

[20, Proposition 2.4] Let R ∈ ℋ and 0 → M ′ → f M → g M ″ → 0 be an exact sequence of R -modules and R -homomorphisms. Then, M is ϕ -torsion if and only if M ′ and M ″ are ϕ -torsion modules.

Proof of Theorem 2.7

Let M be a nonnil-coherent R -module and N be a finitely generated ϕ -submodule of of M . We claim that N is a nonnil-coherent R -module. Let X be a finitely generated ϕ -submodule of N . Then, the following sequence 0 → N / X → M / X → M / N → 0 is exact. Since M / N and N / X are ϕ -torsion modules, so is M / X by Lemma 2.8. Therefore, X is finitely presented, and so N is a nonnil-coherent module.□

Corollary 2.9

If R is a nonnil-coherent ring, then any finitely generated nonnil ideal of R is a nonnil-coherent R-module.

Proof

This follows from Theorem 2.7.□

Theorem 2.10

Let R ∈ ℋ and 0 → P → N → M → 0 be an exact sequence of R-modules and R-homomorphisms, where P is a finitely generated R -module. If N is a nonnil-coherent module, then so is M .

Proof

We can set M = N / P . Let X / P be a finitely generated ϕ -submodule of M . Since N is a nonnil-coherent module and X is a finitely generated ϕ -submodule of N , it follows that X is finitely presented. We claim that X / P is a finitely presented R -module. Actually it follows from [19, (4.54) Lemma] that X / P is finitely presented, and so M is a nonnil-coherent module.□

Corollary 2.11 is a consequence of Theorem 2.10.

Corollary 2.11

Every factor module M / N of a nonnil-coherent module M by a finitely generated submodule N is also a nonnil-coherent module. In particular, every factor module of a nonnil-coherent ring R by a finitely generated ideal I of R is a nonnil-coherent R-module.

Proof

Straightforward.□

Corollary 2.12

Let R ∈ ℋ and M and N be nonnil-coherent modules. Let f : M ⟶ N be an R -homomorphism. Then:

  1. If Im ( f ) is a ϕ -torsion R-module and ker ( f ) is finitely generated, then ker ( f ) is a nonnil-coherent module.

  2. If ker ( f ) is finitely generated, then Im ( f ) is a nonnil-coherent module.

  3. If Coker ( f ) is a ϕ -torsion R-module and Im ( f ) is finitely generated, then Im ( f ) is a nonnil-coherent module.

  4. If Im ( f ) is finitely generated, then Coker ( f ) is a nonnil-coherent module.

Proof

By the following two exact sequences 0 → ker ( f ) → M → Im ( f ) → 0 and 0 → Im ( f ) → N → Coker ( f ) → 0 , the proof is finished using Theorems 2.7 and 2.10.□

Theorem 2.13

Let R ∈ ℋ and 0 → P → f N → g M → 0 be an exact sequence of R -modules and R -homomorphisms. If P and M are nonnil-coherent modules, then so is N.

Proof

Let X be a finitely generated ϕ -submodule of N . Then, we have the following commutative diagram with exact rows:

Since X is a finitely generated module, so is g ( X ) . Let x ∈ M . Then, g ( n ) = x for some n ∈ N . Since N / X is a ϕ -torsion module, s n ∈ X for some s ∈ R ⧹ Nil ( R ) , and so s x ∈ g ( X ) . Therefore, M / g ( X ) is ϕ -torsion. As M is nonnil-coherent, g ( X ) is a finitely presented R -module. Therefore, ker ( g ∣ X ) is a finitely generated R -module since X is finitely generated. Let x ∈ P . Then, there exists t ∈ R ⧹ Nil ( R ) such that t f ( x ) ∈ X , and so t f ( x ) ∈ ker ( g ∣ X ) since g ( t f ( x ) ) = 0 . We can consider f as an embedding, and so P / ker ( g ∣ X ) is a ϕ -torsion module. Then, ker ( g ∣ X ) is finitely presented since P is a nonnil-coherent module, and so X is a finitely presented R -module. Therefore, N is a nonnil-coherent module.□

Corollary 2.14

Let R ∈ ℋ and { M i } i = 1 n be a family of nonnil-coherent modules. Then, ⊕ i = 1 n M i is a nonnil-coherent module.

Proof

We prove this by induction on n . Consider the following exact sequence 0 → M 1 → ⊕ i = 1 n M i → ⊕ i = 2 n M i → 0 and apply Theorem 2.13.□

Corollary 2.15

Let R ∈ ℋ and let M and N be nonnil-coherent submodules of a nonnil-coherent R-module L. If M + N is a ϕ -torsion R-module and M ∩ N is finitely generated, then M + N and M ∩ N are nonnil-coherent modules.

Proof

We use the exact sequence 0 → M ∩ N → M ⊕ N → M + N → 0 and Theorems 2.7 and 2.10.□

Corollary 2.16

Let R ∈ ℋ and I be a finitely generated nonnil ideal of R. Then, R is a nonnil-coherent ring if and only if I and R / I are nonnil-coherent R-modules.

Proof

Assume that R is a nonnil-coherent ring and let I be a finitely generated nonnil ideal of R . By Corollary 2.11, R / I is a nonnil-coherent R -module, and so I is a nonnil-coherent R -module by Theorem 2.7.

Conversely, assume that I and R / I are nonnil-coherent R -modules for any finitely generated nonnil ideal I of R . Then, R is a nonnil-coherent ring by Theorem 2.13.□

Next, Theorem 2.17 gives an analog of the well-known behavior of [5, Theorem 2.2.6].

Theorem 2.17

Let R ∈ ℋ and S be a multiplicative subset of R . If M is a nonnil-coherent R-module, then S − 1 M is a nonnil-coherent ( S − 1 R ) -module.

Proof

It is clear that S − 1 R ∈ ℋ and S − 1 M is a finitely generated ( S − 1 R ) -module. Let N be an R -module such that S − 1 N is a ϕ -torsion ( S − 1 R ) -module. Then, N is a ϕ -torsion R -module. Indeed, let n ∈ N . Then, there exist a ∈ R ⧹ Nil ( R ) and s ∈ S such that a s n 1 = 0 1 . Thus, ( t a ) n = 0 for some t ∈ R ⧹ Nil ( R ) , and so N is a ϕ -torsion R -module. Let X be a finitely generated ( S − 1 R ) -submodule of S − 1 M such that S − 1 M X is ϕ -torsion. Then, we can set X = S − 1 K , where K is a finitely generated submodule of M . Therefore, S − 1 ( M / K ) is ϕ -torsion, and so M / K is a ϕ -torsion R -module. Hence, K is a finitely presented R -module. Thus, X is a finitely presented ( S − 1 R ) -module. Therefore, S − 1 M is a nonnil-coherent ( S − 1 R ) -module.□

Next, we pay attention to the localization of nonnil-coherent rings. Using Theorem 2.17, we obtain immediately:

Corollary 2.18

If R is a nonnil-coherent ring and S is a multiplicative subset of R, then S − 1 R is a nonnil-coherent ring.

Proof

Straightforward.□

Theorem 2.19

Let f : R → T be a finite surjective homomorphism of ϕ -rings (i.e., T is a finitely generated R -module). Let M be a finitely generated T-module which is a nonnil-coherent R-module. Then, M is a nonnil-coherent T-module.

Proof

Let X be a finitely generated T -submodule of M . Then, X is a finitely generated R -module since f is finite. If M / X is a ϕ -torsion T -module, then M / X is a ϕ -torsion R -module, and so X is a finitely presented R -module. Therefore, X is a finitely presented T -module since X ≅ T ⊗ R X . Hence, M is a nonnil-coherent T -module.□

Theorem 2.20

Let R ∈ ℋ and I be a finitely generated nil ideal of R. Let M be an ( R / I ) -module. Then, M is a nonnil-coherent R-module if and only if M is a nonnil-coherent ( R / I ) -module.

In order to prove Theorem 2.20, we need the following lemmas.

Lemma 2.21

[5, Theorem 2.1.8] Let R be a ring and I be a finitely generated ideal of R . Let M be an ( R / I ) -module. Then, M is a finitely presented R-module if and only if M is a finitely presented ( R / I ) -module.

Lemma 2.22

Let R ∈ ℋ and I be a nil ideal of R . Then, R / I ∈ ℋ .

Proof

Note that Nil ( R / I ) = Nil ( R ) / I and R / I Nil ( R / I ) ≅ R / Nil ( R ) is an integral domain, and so Nil ( R / I ) is a prime ideal of R / I . If x ¯ ∈ ( R / I ) ⧹ Nil ( R / I ) , then x ∈ R ⧹ Nil ( R ) , and so Nil ( R ) ⊂ R x . Therefore, Nil ( R / I ) ⊂ ( R / I ) x ¯ , as desired.□

Proof of Theorem 2.20

Assume that M is a nonnil-coherent R -module. Since R / I ∈ ℋ by Lemma 2.22, M is a nonnil-coherent ( R / I ) -module by Theorem 2.19.

Conversely, assume that M is a nonnil-coherent ( R / I ) -module. Then, M is a finitely generated ( R / I ) -module, and so M is a finitely generated R -module. Let X be a finitely generated R -submodule of M such that M / X is a ϕ -torsion R -module. Thus, M / X is a ϕ -torsion ( R / I ) -module, and so X is a finitely presented ( R / I ) -module. By Lemma 2.21, X is a finitely presented R -module. Therefore, M is a nonnil-coherent R -module.□

Corollary 2.23

Let R ∈ ℋ and I be a finitely generated nil ideal of R. Then, R / I is a nonnil-coherent ring if and only if R / I is a nonnil-coherent R-module.

Proof

Straightforward.□

Corollary 2.24

Let R be a nonnil-coherent ring and I be a finitely generated nil ideal of R. Then, R / I is a nonnil-coherent ring.

Proof

This follows immediately from Corollaries 2.11 and 2.23.□

Corollary 2.25

Let R ∈ ℋ and I be a finitely generated nil ideal of R. If R / I is a nonnil-coherent ring and I is a nonnil-coherent R-module, then R is a nonnil-coherent ring.

Proof

This follows directly from Theorem 2.13 and Corollary 2.23.□

3 On nonnil-Noetherian modules

We introduce a new definition of nonnil-Noetherian modules which is different from that in [8]. In [6], an R -module M is said to be Noetherian if every submodule of M is finitely generated.

Definition 3.1

Let R ∈ ℋ . An R -module M is said to be nonnil-Noetherian if every ϕ -submodule of M is finitely generated. In particular, every Noetherian module over a ϕ -ring is nonnil-Noetherian.

Remark 3.2

  1. Note that for a ϕ -torsion R -module M , we have

    M  is nonnil-Noetherian ⇔ M  is Noetherian .

  2. The definition of nonnil-Noetherian modules in Definition 3.1 is different from that of nonnil-Noetherian modules in [8] by Example 2.2 and that of ϕ -Noetherian modules in [9] by Example 2.3. Although the term “non-Noetherian module” used in [8] is the same as in Definition 3.1, we will still use it in the spirit of [7] and following Theorem 3.3.

Recall that in [7], a ϕ -ring R is said to be nonnil-Noetherian if every nonnil ideal of R is finitely generated, equivalently R / Nil ( R ) is a Noetherian domain. The following theorem allows us to see that each nonnil-Noetherian ring is a nonnil-Noetherian module over itself.

Theorem 3.3

Let R be a ϕ -ring. Then, R is a nonnil-Noetherian ring if and only if R is a nonnil-Noetherian module over itself.

Proof

Assume that R is a nonnil-Noetherian ring and let I be an ideal of R such that R / I is ϕ -torsion. Then, I is a nonnil ideal of R , and so I is finitely generated since R is nonnil-Noetherian. Therefore, R is a nonnil-Noetherian module over itself.

Conversely, assume that R is a nonnil-Noetherian module over itself and let I be a nonnil ideal of R . Then, R / I is ϕ -torsion, and so I is finitely generated. Therefore, R is a nonnil-Noetherian ring.□

According to [3, Corollary 3.1], a ϕ -ring R is said to be ϕ -coherent if R / Nil ( R ) is a coherent domain. From [7, Theorem 2.4], a ϕ -ring R is nonnil-Noetherian if and only if R / Nil ( R ) is a Noetherian domain. Therefore, every nonnil-Noetherian ring is ϕ -coherent. The following theorem characterizes when a nonnil-Noetherian ring is nonnil-coherent.

Theorem 3.4

The following statements are equivalent for a nonnil-Noetherian ring R:

  1. R is nonnil-coherent.

  2. R s is a finitely presented ideal of R for any s ∈ R ⧹ Nil ( R ) .

  3. Every nonnil ideal of R is finitely presented.

Proof

( 1 ) ⇒ ( 3 ) ⇒ ( 2 ) They are straightforward.

( 2 ) ⇒ ( 1 ) Assume that R s is finitely presented for every s ∈ R ⧹ Nil ( R ) . Using the exact sequence 0 → ( 0 : s ) → R → R s → 0 , we obtain that ( 0 : s ) is a finitely generated ideal of R . Since R is assumed to be nonnil-Noetherian, and so ϕ -coherent, R is a nonnil-coherent ring by [4, Proposition 1.3].□

Recall that a ϕ -ring R is called a strongly ϕ -ring if Z ( R ) = Nil ( R ) . Strongly ϕ -rings are abundant. Indeed, these rings can be generated from the following pullback introduced by Chang and Kim recently [21]. Let D be a domain with K as its quotient field. Let K [ X ] be the polynomial ring over K , n ≥ 2 be an integer and K [ θ ] = K [ X ] / ⟨ X n ⟩ , where θ ≔ X + ⟨ X n ⟩ . Denote by i : D ↪ K the natural embedding map and π : K [ θ ] ↠ K a ring homomorphism satisfying π ( f ) = f ( 0 ) . Consider the pullback of i and π as follows:

Then, R n = D + θ K [ θ ] = { f ∈ K [ θ ] ∣ f ( 0 ) ∈ D } is a strongly ϕ -ring.

Corollary 3.5

If R is a nonnil-Noetherian strongly ϕ -ring, then R is a nonnil-coherent ring.

Proof

If R is a strongly ϕ -ring, then every principal nonnil ideal is free. Therefore, R is a nonnil-coherent ring if it is nonnil-Noetherian by Theorem 3.4.□

Theorem 3.6

Let 0 → M ′ → f M → g M ″ → 0 be an exact sequence. If M ′ and M ″ are nonnil-Noetherian modules, then so is M. In addition, if M ′ is a ϕ -submodule of M , then the converse holds.

Proof

Assume that M ′ and M ″ are nonnil-Noetherian. Let N be a ϕ -submodule of M . Then, g ( N ) is a ϕ -submodule of M ″ . Indeed, if x ∈ M ″ , then g ( m ) = x for some m ∈ M , and so there exists s ∈ R ⧹ Nil ( R ) such that s m ∈ N . Thus, s x ∈ g ( N ) . Therefore, g ( N ) is a finitely generated submodule of M ″ . Set g ( N ) = ∑ i = 1 t R g ( n i ) , where each n i ∈ N . Let n ∈ N . Then, g ( n ) = ∑ i = 1 t r i g ( n i ) with r i ∈ R . Thus, n − ∑ i = 1 t r i n i ∈ ker ( g ) = Im ( f ) , and so n = f ( y ) + ∑ i = 1 t r i n i for some y ∈ M ′ . In addition, M ′ is finitely generated since it is nonnil-Noetherian. Thus, M ′ = ∑ i = t + 1 t + l R n i for some n t + 1 , n t + 2 , … , n t + l ∈ M , and so there exists r t + 1 , r t + 2 , … , r t + l ∈ R such that f ( y ) = ∑ i = t + 1 t + l r i n i . Hence, n = ∑ i = 1 t + l r i n i . Therefore, N is finitely generated, and so M is a nonnil-Noetherian module.

Assume that M is a nonnil-Noetherian module and M ′ is a ϕ -submodule of M . Let X be a ϕ -submodule of M ′ . Then, 0 → M ′ / X → M / X → M ″ → 0 is exact with M ′ / X and M ″ ϕ -torsion, and so X is a ϕ -submodule of M . Thus, X is a finitely generated submodule of M ′ . Therefore, M ′ is a nonnil-Noetherian module. Let N be a submodule of M such that M ′ ⊂ N and N / M ′ is a ϕ -submodule of M ″ ≅ M / M ′ . We claim that N is a ϕ -submodule of M . If x ∈ M , then s ( x + M ′ ) = M ′ for some s ∈ R ⧹ Nil ( R ) since M / M ′ is ϕ -torsion, and so s x ∈ M ′ ⊂ N . Thus, N is a ϕ -submodule of M . Therefore, N is a finitely generated submodule of M , and so N / M ′ is a finitely generated submodule of M ″ . Therefore, M ″ is a nonnil-Noetherian module.□

Corollary 3.7

Let R ∈ ℋ and M be a nonnil-Noetherian R-module. Then, every ϕ -submodule of M is nonnil-Noetherian.

Proof

This follows immediately from Theorem 3.6.□

Corollary 3.8

Let R ∈ ℋ and { M i } 1 ≤ i ≤ n be a family of nonnil-Noetherian modules. Then, ⊕ i = 1 n M i is a nonnil-Noetherian module.

Proof

We prove this by induction on n . Consider the following exact sequence 0 → M 1 → ⊕ i = 1 n M i → ⊕ i = 2 n M i → 0 and apply Theorem 3.6.□

Corollary 3.9

If R is a nonnil-Noetherian ring, then every finitely generated ϕ -torsion module is nonnil-Noetherian (and so is Noetherian).

Proof

If M is a finitely generated ϕ -torsion R -module, then M ≅ R ( n ) / N , where n ∈ N and N is a submodule of R ( n ) . Since M is ϕ -torsion, N is a ϕ -submodule of R ( n ) . Using the exact sequence 0 → N → R ( n ) → M → 0 and Theorem 3.6, we can deduce that M is nonnil-Noetherian.□

Corollary 3.10

Let R ∈ ℋ and I be a finitely generated nonnil ideal of R. Then, R is a nonnil-Noetherian ring if and only if I and R / I are nonnil-Noetherian R-modules.

Proof

This follows immediately from Theorem 3.6.□

Theorem 3.11

Let R ∈ ℋ . If M is a nonnil-Noetherian R-module, then every factor module of M is nonnil-Noetherian.

Proof

Let M be a nonnil-Noetherian module and N be a submodule of M . We claim that M / N is a nonnil-Noetherian module. Let P / N be a ϕ -submodule of M / N , where P is a submodule of M containing N . Since M / N P / N ≅ M P is a ϕ -torsion R -module, P is finitely generated, and so P / N is a finitely generated submodule of M / N . Therefore, M / N is nonnil-Noetherian.□

Corollary 3.12

If R is a nonnil-Noetherian ring and I is an ideal of R, then R / I is a nonnil-Noetherian R-module.

Proof

This follows immediately from Theorem 3.11.□

Corollary 3.13

Let R be a nonnil-Noetherian ring and M be an R-module. Then, M is a nonnil-Noetherian module if and only if M is a finitely generated R-module.

Proof

If M is a nonnil-Noetherian module, then it is easy to see that M is a finitely generated module. Conversely, if M is a finitely generated module, then M is a factor of R ( n ) , where n ∈ N . Since R ( n ) is a nonnil-Noetherian module by Corollary 3.8, M is a nonnil-Noetherian module by Theorem 3.11.□

Corollary 3.14

A ring R is nonnil-Noetherian if and only if every ϕ -submodule of a finitely generated R-module is finitely generated.

Proof

Straightforward.□

Theorem 3.15 establishes that every finitely generated ϕ -torsion module over a nonnil-Noetherian ring is finitely presented.

Theorem 3.15

Let R be a nonnil-Noetherian ring and M be a finitely generated ϕ -torsion R-module. Then, M is finitely presented.

Proof

Let M be a finitely generated ϕ -torsion R -module. Then, there exist n ∈ N and a sequence 0 → N → R ( n ) → M → 0 . Since R ( n ) is a nonnil-Noetherian R -module by Corollary 3.8 and M is a ϕ -torsion module, N is a finitely generated module. Therefore, M is a finitely presented module.□

Theorem 3.16 establishes that the class of nonnil-Noetherian modules is closed under localizations.

Theorem 3.16

Let R be a ϕ -ring and S be a multiplicative subset of R. If M is a nonnil-Noetherian R-module, then S − 1 M is a nonnil-Noetherian ( S − 1 R ) -module.

Proof

Let M be a nonnil-Noetherian R -module and S − 1 N be a ϕ -submodule of S − 1 M , where N is a submodule of M . Then, N is a ϕ -submodule of M , and so N is a finitely generated R -module. Thus, S − 1 N is a finitely generated ( S − 1 R ) -module. Therefore, S − 1 M is a nonnil-Noetherian ( S − 1 R ) -module.□

Corollary 3.17

If R is a nonnil-Noetherian ring and S is a multiplicative subset of R, then S − 1 R is a nonnil-Noetherian ring.

Proof

This follows immediately from Theorem 3.16.□

We end this section by the following theorem.

Theorem 3.18

Let R be a nonnil-Noetherian ring and I be a nil ideal of R. Then, R/I is a nonnil-Noetherian ring.

Proof

Let J / I be a nonnil ideal of R / I . Then, R / I J / I ≅ R / J is a ϕ -torsion R -module, and so J is a nonnil ideal of R . As R is nonnil-Noetherian, J is a finitely generated ideal of R , and so J / I is a finitely generated ideal of R / I . Therefore, R / I is nonnil-Noetherian.□

4 Transfer of nonnil-coherence and nonnil-Noetherianity in trivial ring extensions

Now, we study the transfer of nonnil-coherent rings in the trivial ring extensions. From [1, Corollary 2.4], a trivial ring extension R ∝ M is a ϕ -ring if and only if R is a ϕ -ring and M is a ϕ -divisible module (i.e., s M = M for all s ∈ R ⧹ Nil ( R ) ).

Let M be an R -module and r ∈ R . Set ( 0 : M r ) ≔ { m ∈ M ∣ r m = 0 } . It is easy to verify that ( 0 : M r ) is a submodule of M such that ( 0 : r ) M ⊂ ( 0 : M r ) . Therefore, ( 0 : r ) ∝ ( 0 : M r ) is an ideal of R ∝ M by [22, Theorem 3.1].

The following theorem characterizes when a trivial ring extension is a nonnil-coherent ring.

Theorem 4.1

Let A ∈ ℋ , M be a ϕ -divisible A-module, and set R ≔ A ∝ M . Then, the following statements are equivalent:

  1. R is a nonnil-coherent ring.

  2. A is a nonnil-coherent ring and ( 0 : r ) ∝ ( 0 : M r ) is a finitely generated ideal of R for each r ∈ A ⧹ Nil ( A ) .

  3. A is a nonnil-coherent ring and R ( r , 0 ) is finitely presented for all r ∈ A ⧹ Nil ( A ) .

Before proving Theorem 4.1, we need the following lemmas:

Lemma 4.2

Let A ∈ ℋ and M be a ϕ -divisible A-module. Let J be an ideal of R ≔ A ∝ M . Then, J is a nonnil ideal of R if and only if there exists a unique nonnil ideal I of A such that J = I ∝ M .

Proof

Assume that J is a nonnil ideal of R . Then, 0 ∝ M ⊂ Nil ( R ∝ M ) ⊂ J , and so J = I ∝ M for a unique nonnil ideal I of R by [22, Theorem 3.1].

Conversely, assume that J = I ∝ M for a unique nonnil ideal I of A . Then, it is clear that J is a nonnil ideal of R .□

Lemma 4.3

Let A ∈ ℋ and M be a ϕ -divisible A-module. Let J = I ∝ M be a nonnil ideal of R = A ∝ M . Then, J is a finitely generated nonnil ideal of R if and only if I is a finitely generated nonnil ideal of A.

Proof

Assume that I is a finitely generated nonnil ideal of A . Then, I = ∑ i = 1 n A a i , where each a i ∈ A , and we may assume that a 1 ∈ A ⧹ Nil ( A ) . First, it is easy to see that ∑ i = 1 n R ( a i , 0 ) ⊂ J . Conversely, let ( α , β ) ∈ J . Then α = ∑ i = 1 n r i a i for some r i ∈ A . Since M is ϕ -divisible, β = a 1 v 1 for some v 1 ∈ M , and so ( α , β ) = ∑ i = 1 n ( a i , 0 ) ( r i , v i ) , where v i = 0 for all 2 ≤ i ≤ n . Therefore, J ⊂ ∑ i = 1 n R ( a i , 0 ) , and so J = ∑ i = 1 n R ( a i , 0 ) is a finitely generated nonnil ideal. The converse is straightforward.□

Lemma 4.4

Let A ∈ ℋ and M be a ϕ -divisible A-module. Let r be a non-nilpotent element of A and u ∈ M . Then,

( ( 0 , 0 ) : ( r , u ) ) = ( 0 : r ) ∝ ( 0 : M r ) .

Proof

Let ( r , u ) ∈ A ⧹ Nil ( A ) ∝ M and ( α , β ) ∈ ( ( 0 , 0 ) : ( r , u ) ) . Since M is ϕ -divisible, u = r v for some v ∈ M , and so ( r , u ) = ( r , 0 ) ( 1 , v )

( α , β ) ∈ ( ( 0 , 0 ) : ( r , u ) ) ⇔ ( α , β ) ( r , u ) = ( 0 , 0 ) ⇔ ( α , β ) ( r , 0 ) ( 1 , v ) = ( 0 , 0 ) ⇔ ( α r , α r v + β r ) = ( 0 , 0 ) ⇔ ( α , β ) ∈ ( 0 : r ) ∝ ( 0 : M r ) .

Therefore, ( ( 0 , 0 ) : ( r , u ) ) = ( 0 : r ) ∝ ( 0 : M r ) . □

Lemma 4.5

[3, Theorem 2.1] A ϕ -ring R is nonnil-coherent if and only if ( 0 : r ) is a finitely generated ideal for every non-nilpotent element r ∈ R , and the intersection of two finitely generated nonnil ideals of R is a finitely generated nonnil ideal of R.

Proof of Theorem 4.1

( 1 ) ⇒ ( 2 ) Assume that R is a nonnil-coherent ring. Let I and J be finitely generated nonnil ideals of A . Then, I ∝ M and J ∝ M are finitely generated nonnil ideals of R by Lemma 4.3. Since R is a nonnil-coherent ring, ( I ∝ M ) ∩ ( J ∝ M ) = ( I ∩ J ) ∝ M is a finitely generated nonnil ideal of R by Lemma 4.5. Therefore, I ∩ J is a finitely generated nonnil ideal of A by Lemma 4.3. Let r ∈ A ⧹ Nil ( A ) . Then, ( 0 : r ) ∝ ( 0 : M r ) is a finitely generated ideal of R by Lemma 4.4, and so ( 0 : r ) is a finitely generated ideal of A . Therefore, A is a nonnil-coherent ring by Lemma 4.5.

( 2 ) ⇒ ( 1 ) Assume that A is a nonnil-coherent ring and ( 0 : r ) ∝ ( 0 : M r ) is a finitely generated ideal of R for each r ∈ A ⧹ Nil ( A ) . Let I ∝ M and J ∝ M be finitely generated nonnil ideals of R . Then, I and J are finitely generated nonnil ideals of A . Since A is a nonnil-coherent ring, I ∩ J is a finitely generated nonnil ideal of A , and so ( I ∝ M ) ∩ ( J ∝ M ) = ( I ∩ J ) ∝ M is a finitely generated nonnil ideal of R by Lemma 4.3. Let ( r , u ) ∈ R ⧹ Nil ( R ) . Then, ( ( 0 , 0 ) : ( r , u ) ) = ( 0 : r ) ∝ ( 0 : M r ) is a finitely generated ideal of R by hypothesis. Therefore, R is a nonnil-coherent ring by Lemma 4.5.

( 2 ) ⇔ ( 3 ) Let r ∈ A ⧹ Nil ( A ) and u ∈ M . Then, the following sequence 0 → ( ( 0 , 0 ) : ( r , u ) ) → R → R ( r , 0 ) → 0 is exact. Therefore, by Lemma 4.4, ( 0 : r ) ∝ ( 0 : M r ) is a finitely generated ideal of R if and only if R ( r , 0 ) is finitely presented.□

Corollary 4.6

Let R = A ∝ M be a ϕ -ring such that Z ( A ) = Nil ( A ) . Then, R is a nonnil-coherent ring if and only if A is a nonnil-coherent ring and ( 0 : M r ) is a finitely generated A -submodule of M for every r ∈ A ⧹ Nil ( A ) .

Proof

Let r ∈ A ⧹ Nil ( A ) . Since Z ( A ) = Nil ( A ) , it follows that ( 0 : r ) = 0 . Therefore, ( ( 0 , 0 ) : ( r , u ) ) = 0 ∝ ( 0 : M r ) . Now the assertion follows immediately from Theorem 4.1.□

Corollary 4.7

Let R = A ∝ M be a ϕ -ring such that Z ( A ) = Nil ( A ) and M is a Noetherian A-module. Then, R is a nonnil-coherent ring if and only if A is a nonnil-coherent ring.

Proof

This follows immediately from Theorem 4.1.□

For a ring R and an R -module M , set Z R ( M ) ≔ { r ∈ R ∣ r m = 0 for some nonzero m ∈ M } .

Corollary 4.8

Let R = A ∝ M be a ϕ -ring such that Z ( A ) = Nil ( A ) = Z A ( M ) . Then, R is a nonnil-coherent ring if and only if A is a nonnil-coherent ring.

Proof

It is easy to see that ( 0 : r ) = 0 and ( 0 : M r ) = 0 for each r ∈ A ⧹ Nil ( A ) . Now the proof follows directly from Theorem 4.1.□

Example 4.9

  1. Z ∝ Q is a nonnil-coherent ring.

  2. Z / 4 Z ∝ Z / 2 Z is a nonnil-coherent ring.

The following theorem studies the transfer of being a ϕ -coherent ring in trivial extensions.

Theorem 4.10

Let A ∈ ℋ and M be a ϕ -divisible A-module. Then, A ∝ M is a ϕ -coherent ring if and only if A is a ϕ -coherent ring.

Proof

First, note that Nil ( A ∝ M ) = Nil ( A ) ∝ M , and so A ∝ M Nil ( A ∝ M ) ≅ A / Nil ( A ) . Therefore, A ∝ M is a ϕ -coherent ring if and only if A is a ϕ -coherent ring.□

Recently, Qi and Zhang [4] provided for the first time an example of a ϕ -coherent ring, which is not nonnil-coherent. Now, we give a concrete example by using Corollary 4.6 and Theorem 4.10.

Example 4.11

Let E = ⊕ i = 1 ∞ Q / Z . Then, E is a divisible abelian group. Therefore, R = Z ∝ E is a ϕ -ring. Since

( 0 : E 2 ) = a i b i + Z i ∈ N ∗ ∣ a i ∈ Z and g c d ( a i , b i ) = 1 , b i ∈ { 1 , 2 } ∀ i ∈ N ∗ ,

which is an infinitely generated abelian group. Therefore, R is not a nonnil-coherent ring by Corollary 4.6. Note that R is an example of a ϕ -coherent ring, which is not nonnil-coherent by Theorem 4.10.

Now, we study the transfer of nonnil-Noetherian rings in the trivial ring extensions.

Theorem 4.12

Let A ∈ ℋ and M be a ϕ -divisible R-module. Then, A ∝ M is a nonnil-Noetherian ring if and only if A is a nonnil-Noetherian ring.

Proof

A ∝ M is nonnil-Noetherian ring if and only if A ∝ M Nil ( A ) ∝ M ≅ A Nil ( A ) is a Noetherian domain and A is a nonnil-Noetherian ring.□

We give some examples of nonnil-Noetherian extension rings A ∝ M that are nonnil-coherent.

Example 4.13

If R = A ∝ M is a ϕ -ring such that Z ( A ) = Nil ( A ) = Z A ( M ) , then for all r ∈ A ⧹ Nil ( A ) , we obtain ( 0 : r ) = 0 and ( 0 : M r ) = 0 , and so ( A ∝ M ) ( r , 0 ) ≅ A ∝ M . Therefore, it follows from Theorem 3.4 that A ∝ M is nonnil-coherent if it is nonnil-Noetherian.

Example 4.14

Let A be a strongly ϕ -ring and M be a ϕ -torsion-free A -module. If A ∝ M is a nonnil-Noetherian ring, then A ∝ M is a nonnil-coherent ring.

Proof

Let ( α , m ) ∈ A ∝ M such that ( α , m ) ( r , 0 ) = ( 0 , 0 ) . Then, α r = 0 , and r m = 0 and so ( α , m ) = ( 0 , 0 ) . Thus, ( A ∝ M ) ( r , 0 ) is a finitely generated free ideal. Hence, if A ∝ M is a nonnil-Noetherian ring, then A ∝ M is a nonnil-coherent ring by Theorem 3.4.□

Recall that every nonnil-Noetherian ring is ϕ -coherent. The following Example 4.15 gives a ϕ -coherent ring that is not nonnil-Noetherian.

Example 4.15

Let R ≔ ( Z + X Q [ [ X ] ] ) ∝ q f ( Q [ [ X ] ] ) . Then, R is a ϕ -coherent ring that is not nonnil-Noetherian.

Proof

First, it is easy to see that R is a ϕ -ring by [1, Corollary 2.4]. By [23, Theorem 3], Z + X Q [ [ X ] ] is a coherent domain, and so R is a ϕ -coherent ring by Theorem 4.10. By [23, Theorem 3], Z + X Q [ [ X ] ] is not a Noetherian domain, and so is not nonnil-Noetherian. Therefore, R is never a nonnil-Noetherian ring by Theorem 4.12.□

Remark 4.16

Note that the ring R in Example 4.15 is nonnil-coherent since R is a strongly ϕ -ring, and a ϕ -ring A is nonnil-coherent if and only if A is ϕ -coherent and ( 0 : a ) is a finitely generated ideal of A for each a ∈ A ⧹ Nil ( A ) ([4, Proposition 1.3]). Using Examples 4.11 and 4.15, we can deduce that the converse of the following implications are not true in general:

nonnil-Noetherian ⇒ nonnil-coherent ⇒ ϕ -coherent .

5 On transfer nonnil-Noetherian and ϕ -coherent rings in the amalgamation algebra along an ideal

In this section, we study the transfer of nonnil-Noetherian rings in the amalgamation algebra along an ideal. El Khalfi et al. [1] studied when the amalgamation algebra along an ideal is a ϕ -ring, a ϕ -chained ring, and a ϕ -pseudo-valuation ring.

Our next result characterizes when the amalgamation of a ring is a nonnil-Noetherian ring. Before starting this section, we need the following theorems.

Theorem 5.1

[1, Proposition 2.20] Let f : A → B be a ring homomorphism and J be an ideal of B . Then,

Nil ( A ⋈ f J ) = { ( a , f ( a ) + j ) ∣ a ∈ Nil ( A ) and j ∈ J ∩ Nil ( B ) } .

Theorem 5.2

[1, Theorem 2.1] Let f : A → B be a ring homomorphism and J be a nonnil ideal of B. Set N ( J ) ≔ J ∩ Nil ( B ) . The following statements are equivalent:

  1. R = A ⋈ f J ∈ ℋ .

  2. A is an integral domain, f − 1 ( J ) = 0 , and N ( J ) is a divided prime ideal of f ( A ) + J .

Theorem 5.3 studies the transfer of being a nonnil-Noetherian ring between a ϕ -ring A and an amalgamation algebra A ⋈ f J along a nonnil ideal J .

Theorem 5.3

Let A and B be two rings and f : A ⟶ B be a ring homomorphism. Let J be a nonnil ideal of B. Define f ¯ : A ⟶ B / N ( J ) by f ¯ ( a ) = f ( a ) + N ( J ) for all a ∈ A . If A ⋈ f J is a ϕ -ring, then the following statements are equivalent:

  1. A ⋈ f J is a nonnil-Noetherian ring.

  2. A ⋈ f ¯ J N ( J ) is a Noetherian domain.

  3. f − 1 ( J ) = { 0 } , A , and f ¯ ( A ) + J / N ( J ) are Noetherian domains.

Before proving Theorem 5.3, we establish the following lemmas.

Lemma 5.4

With the notations of Theorem 5.3, we obtain f ¯ − 1 ( J / N ( J ) ) = f − 1 ( J ) .

Proof

Straightforward.□

Lemma 5.5

Let f : A ⟶ B be a ring homomorphism and J be a nonzero ideal of B. Let J ′ be a subideal of J and I be an ideal of A such that f ( I ) ⊂ J ′ . Define f ¯ ¯ : A / I ⟶ B / J ′ by f ¯ ¯ ( a ¯ ) = f ( a ) ¯ , where a ¯ ≔ a + I and f ( a ) ¯ ≔ f ( a ) + J ′ . Then, we have the following ring isomorphism:

A ⋈ f J I ⋈ f J ′ ≅ A I ⋈ f ¯ ¯ J J ′ .

Proof

Define

φ : A ⋈ f J ⟶ A I ⋈ f ¯ ¯ J J ′ ( a , f ( a ) + j ) ⟼ ( a ¯ , f ( a ) ¯ + j ¯ ) .

It is easy to see that φ is a surjective ring homomorphism and for all ( a , f ( a ) + j ) ∈ A ⋈ f J , ( a ¯ , f ( a ) ¯ + j ¯ ) = ( 0 ¯ , 0 ¯ ) if and only if a ∈ I and j ∈ J ′ and ( a , f ( a ) + j ) ∈ I ⋈ f J ′ . Therefore A ⋈ f J I ⋈ f J ′ ≅ A I ⋈ f ¯ ¯ J J ′ .□

Proof of Theorem 5.3

( 1 ) ⇒ ( 2 ) Assume that A ⋈ f J is a nonnil-Noetherian ring. Since A ⋈ f J ∈ ℋ , A is an integral domain by Theorem 5.2. Therefore, Nil ( A ⋈ f J ) = 0 × N ( J ) . As A ⋈ f J is a nonnil-Noetherian ring, A ⋈ f J 0 × N ( J ) is a Noetherian domain. Therefore, A ⋈ f ¯ J N ( J ) is a Noetherian domain by Lemma 5.5.

( 2 ) ⇒ ( 1 ) This follows immediately from Lemma 5.5.

( 2 ) ⇒ ( 3 ) Assume that A ⋈ f ¯ J / N ( J ) is a Noetherian domain. By [13, Proposition 5.2] and Lemma 5.4, f − 1 ( J ) = 0 and f ¯ ( A ) + J / N ( J ) is an integral domain. By [13, Proposition 5.6], A and f ¯ ( A ) + J / N ( J ) are Noetherian domains, as desired.

( 3 ) ⇒ ( 2 ) By Lemma 5.4, we have f ¯ − 1 ( J / N ( J ) ) = 0 . By [13, Proposition 5.1], f ¯ ( A ) + J / N ( J ) ≅ A ⋈ f ¯ J / N ( J ) , which is a Noetherian domain, as desired.□

Recall from [1, Corollary 2.6] that a polynomial ring R [ X ] is a ϕ -ring if and only if R is an integral domain.

Theorem 5.6

Let R be an integral domain. Then, R [ X ] is a nonnil-Noetherian ring if and only if R [ X ] is a Noetherian domain.

Proof

By [1, Corollary 2.6], R [ X ] is a ϕ -ring and R [ X ] ≅ R ⋈ j J , where J = X R [ X ] and j : R ↪ R [ X ] . Since J ⊄ Nil ( R [ X ] ) , it follows that R [ X ] is a nonnil-Noetherian ring if and only if R ⋈ j J is a Noetherian domain by Theorem 5.3.□

Corollary 5.7

Let A be a ring and J be a nonnil ideal of A. Assume that A ⋈ J ∈ ℋ . Then, A ⋈ J is never a nonnil-Noetherian ring.

Proof

Assume, on the contrary, that A ⋈ J is a nonnil-Noetherian ring. Then, A ⋈ J / N ( J ) is a Noetherian domain, and so A is a Noetherian domain with J = N ( J ) by [13, Remark 5.3]. Therefore, J ⊂ Nil ( A ) , a desired contradiction.□

Theorem 5.8 studies the transfer of being a nonnil-Noetherian ring between a ϕ -ring A and an amalgamation algebra A ⋈ f J along a nil ideal J .

Theorem 5.8

Let A and B be two rings and f : A ⟶ B be a ring homomorphism. Let J be a nil ideal of B. Assume that A ⋈ f J is a ϕ -ring. Then, A ⋈ f J is a nonnil-Noetherian ring if and only if A is a nonnil-Noetherian ring.

Proof

Note that J ⊂ Nil ( B ) , and thus N ( J ) = J . So Nil ( A ⋈ f J ) = Nil ( A ) ⋈ f J . Therefore, A ⋈ f J is a nonnil-Noetherian ring if and only if A ⋈ f J Nil ( A ) ⋈ f J is a Noetherian domain, and A Nil ( A ) is a Noetherian domain, and A is a nonnil-Noetherian ring.□

Example 5.9

R = Z [ X ] ∝ q f ( Z [ X ] ) is a nonnil-Noetherian ring that is not a Noetherian ring.

Now, we study the transfer of being ϕ -coherent rings in the amalgamation algebra along an ideal.

Theorem 5.10

Let A and B be two rings and f : A ⟶ B be a ring homomorphism. Let J be a nonnil ideal of B. Define f ¯ : A ⟶ B / N ( J ) by f ¯ ( a ) = f ( a ) + N ( J ) for any a ∈ A . Assume that A ⋈ f J is a ϕ -ring. Then, the following statements are equivalent:

  1. A ⋈ f J is a ϕ -coherent ring.

  2. A ⋈ f ¯ J N ( J ) is a coherent domain.

  3. f − 1 ( J ) = { 0 } and f ¯ ( A ) + J / N ( J ) is a coherent domain.

Proof

( 1 ) ⇒ ( 2 ) Assume that A ⋈ f J is a ϕ -coherent ring. Since A ⋈ f J ∈ ℋ , it follows that A is an integral domain by Theorem 5.2, and so Nil ( A ⋈ f J ) = 0 × N ( J ) . As A ⋈ f J is a ϕ -coherent ring, A ⋈ f J 0 × N ( J ) is a coherent domain. Therefore, A ⋈ f ¯ J N ( J ) is a coherent domain by Lemma 5.5.

( 2 ) ⇒ ( 1 ) This follows directly from Lemma 5.5.

( 2 ) ⇒ ( 3 ) Assume that A ⋈ f ¯ J / N ( J ) is a coherent domain. From [13, Proposition 5.2] and Lemma 5.4, f − 1 ( J ) = 0 and f ¯ ( A ) + J / N ( J ) is an integral domain. From [13, Proposition 5.1], f ¯ ( A ) + J / N ( J ) ≅ A ⋈ f ¯ J / N ( J ) , as desired.

( 3 ) ⇒ ( 2 ) By Lemma 5.4 we have f ¯ − 1 ( J / N ( J ) ) = 0 and from [13, Proposition 5.1], we obtain f ¯ ( A ) + J / N ( J ) ≅ A ⋈ f ¯ J / N ( J ) , which is a coherent domain, as desired.□

Corollary 5.11

Let R be an integral domain. Then, R [ X ] is a ϕ -coherent ring if and only if R [ X ] is a coherent domain.

Proof

By [1, Corollary 2.6], we have that R [ X ] is a ϕ -ring and R [ X ] ≅ R ⋈ j J , where J = X R [ X ] and j : R ↪ R [ X ] . Since J ⊄ Nil ( R [ X ] ) , it follows that R [ X ] is a ϕ -coherent ring if and only if R ⋈ j J is a coherent domain by Theorem 5.10.□

Corollary 5.12 studies the transfer of being a nonnil-coherent ring between a ϕ -ring A and an amalgamation algebra A ⋈ f J along a nonnil ideal J .

Corollary 5.12

Let A and B be two rings and f : A ⟶ B be a ring homomorphism. Let J be a nonnil ideal of B . Define f ¯ : A ⟶ B / N ( J ) by f ¯ ( a ) = f ( a ) + N ( J ) for any a ∈ A . Assume A ⋈ f J is a ϕ -ring. Then, the following statements are equivalent:

  1. A ⋈ f J is a nonnil-coherent ring,

  2. The following conditions hold:

    1. f − 1 ( J ) = { 0 } .

    2. f ¯ ( A ) + J / N ( J ) is a coherent domain.

    3. ( A ⋈ f J ) ( r , f ( r ) + j ) is a finitely presented ideal for any non-nilpotent element ( r , f ( r ) + j ) of A ⋈ f J .

Proof

This follows immediately from [4, Proposition 1.3] and Theorem 5.10□

Theorem 5.13 studies the transfer of being a ϕ -coherent ring between a ϕ -ring A and an amalgamation algebra A ⋈ f J along a nil ideal J .

Theorem 5.13

Let A and B be two rings and f : A ⟶ B be a ring homomorphism. Let J be a nil ideal of B. Assume that A ⋈ f J is a ϕ -ring. Then, A ⋈ f J is a ϕ -coherent ring if and only if A is a ϕ -coherent ring.

Proof

Since J ⊂ Nil ( B ) , we have N ( J ) = J . It is easy to see that Nil ( A ⋈ f J ) = Nil ( A ) ⋈ f J . Therefore, A ⋈ f J is a ϕ -coherent ring, A ⋈ f J Nil ( A ) ⋈ f J is a coherent domain, A Nil ( A ) is a coherent domain, and A is a ϕ -coherent ring.□

Corollary 5.14 studies the transfer of being a nonnil-coherent ring between a ϕ -ring A and an amalgamation algebra A ⋈ f J along a nil ideal J .

Corollary 5.14

Let A and B be two rings and f : A ⟶ B be a ring homomorphism. Let J be a nil ideal of B. Assume that A ⋈ f J is a ϕ -ring. Then, the following are equivalent:

  1. A ⋈ f J is a nonnil-coherent ring.

  2. A is a ϕ -coherent ring and ( A ⋈ f J ) ( r , f ( r ) + j ) is a finitely presented ideal for any non-nilpotent element ( r , f ( r ) + j ) of A ⋈ f J .

Proof

This follows immediately from [4, Proposition 1.3] and Theorem 5.13.□

Acknowledgements

We would like to thank the reviewers for their valuable comments and suggestions that significantly improved our manuscript.

  1. Funding information: H.K. was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2021R1I1A3047469).

  2. Author contributions: Y.E.H. and N.M. conceived of the presented idea. All authors developed the theory, discussed the results, contributed to the final manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-05-27
Revised: 2022-10-01
Accepted: 2022-10-28
Published Online: 2022-11-24

© 2022 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0526
Keywords for this article
nonnil-coherent ring; ϕ-coherent ring; ϕ-submodule; nonnil-coherent module; nonnil-Noetherian ring; nonnil-Noetherian module
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  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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