About j-Noetherian rings

Khaled Alhazmy; Fuad Ali Ahmed Almahdi; Najib Mahdou; El Houssaine Oubouhou

doi:10.1515/math-2024-0014

Article Open Access

About j-Noetherian rings

Khaled Alhazmy , Fuad Ali Ahmed Almahdi , Najib Mahdou and El Houssaine Oubouhou

Published/Copyright: May 17, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

Let R be a commutative ring with identity and j an ideal of R . An ideal I of R is said to be a j -ideal if I ⊈ j . We define R to be a j -Noetherian ring if each j -ideal of R is finitely generated. In this work, we study some properties of j -Noetherian rings. More precisely, we investigate j -Noetherian rings via the Cohen-type theorem, the flat extension, decomposable ring, the trivial extension ring, the amalgamated duplication, the polynomial ring extension, and the power series ring extension.

Keywords: 𝒿-Noetherian rings; 𝒿-ideals; nonnil-Noetherian rings

MSC 2010: 13Axx; 13Bxx; 13Cxx; 13E99

1 Introduction

Throughout this study, it is assumed that all rings are commutative with non-zero identity. The concept of Noetherian rings is one of the most important topics that is widely used in many areas including commutative algebra and algebraic geometry. The Noetherian property was originally initiated by the mathematician Noether who first considered a relation between the ascending chain condition on ideals and the finite generation of ideals. More precisely, she showed that if R is a ring, then the ascending chain condition on ideals of R holds if and only if every ideal of R is finitely generated. The equivalence plays a significant role in simplifying the ideal structure of a ring. Due to the importance of Noetherian rings, many mathematicians have tried to use Noetherian properties in several classes of rings and attempted to generalize the notion of Noetherian rings. Nonnil-Noetherian rings and S -Noetherian rings are typical generalizations of Noetherian rings. The reader can refer to [1–4] for nonnil-Noetherian rings, and to [5–8] for S -Noetherian rings.

Let R be a commutative ring, we denote by Nil ( R ) the ideal of all nilpotent elements of R . Recall that an ideal I of R is said to be a nonnil ideal if I ⊈ Nil ( R ) . We say that R is nonnil-Noetherian if each nonnil ideal of R is finitely generated. This notion was introduced and studied by Badawi in 2003 under the strong hypothesis that the nilradical of the ring is a divided prime ideal. Later, Hizem and Benhissi in 2011 extended some properties of nonnil-Noetherian rings without any assumption on the nilradical. It is worth mentioning that nonnil-Noetherian rings have their own benefits. For example, in nonnil-Noetherian rings, we can investigate non-prime ideals because the nilradical is the intersection of prime ideals of a ring. Many features of Noetherian rings are analogously proved for Nonnil-Noetherian rings. In [1], the trivial extension construction is provided to give examples of nonnil-Noetherian rings which are not Noetherian rings.

The main purpose of this work is to introduce and investigate the notion of j -Noetherian rings which is a generalization of nonnil-Noetherian rings and Noetherian rings. Let R be a ring and j an ideal of R . Set R ( j ) = { I ∣ I is an ideal of R such that I ⊈ j } . If I ∈ R ( j ) , then I is called a j -ideal. In particular, if j = 0 , then every non-zero ideal is a j -ideal, and if j = Nil ( R ) , then the notion of j -ideal coincides with the notion of nonnil ideal. Clearly, if j 1 ⊂ j 2 are two ideals of R , then every j 2 -ideal is j 1 -ideal. We define R to be a j -Noetherian ring if each j -ideal of R is finitely generated. Note that if ( R , j ) is a local ring, then R is always a j -Noetherian ring, and if j = 0 , then the notion of j -Noetherian rings coincides with that of Noetherian ring. Furthermore, if j = Nil ( R ) , then the concept of j -Noetherian rings is precisely the same as that of nonnil-Noetherian rings. Clearly, if j 1 ⊆ j 2 are ideals of a ring R and R is j 1 -Noetherian, then it is a j 2 -Noetherian ring. In this article, we establish the Eakin-Nagata-Formanek theorem, the Cohen-type theorem, and the flat extension of j -Noetherian rings. After that, we investigate some ring extensions of j -Noetherian rings. More precisely, we study the amalgamated duplication, the trivial ring extension construction, polynomial ring extension, and the power series ring extension of j -Noetherian rings. We also study some properties of polynomial ring and power series over j -Noetherian rings. For any undefined terminology and notation, the reader is referred to [9–12].

2 Results

The first result of this study provides the Eakin-Nagata-Formanek theorem for j -Noetherian rings.

Theorem 2.1

Let R be a ring and j a proper ideal of R. Then, the following statements are equivalent.

Every nonempty family of j -ideals has a maximal element.
R is j -Noetherian.
Every ascending chain of j -ideals of R is stationary.
For every j -ideal I of R , R ∕ I is a Noetherian ring.

Proof

( 1 ) ⇒ ( 2 ) Let I be a j -ideal of R . Set Ω to be the set of all finitely generated j -ideals of R which are included in I . Since I is a j -ideal of R , there exists a ∈ I \ j . This makes a R ∈ Ω , and hence Ω is nonempty. By the assumption, Ω has a maximal element L . On the other hand, L is finitely generated. Write L = x 1 R + … + x n R . Now, our aim is to prove that I = L . Suppose that I ≠ L , so there exists α ∈ I such that α ∉ L . Set Q = L + α R . Therefore, Q ⊆ I and Q is a finitely generated j -ideal of R and so Q ∈ Ω . Since L ⊆ Q , by the maximality of L , Q ⊆ L . Therefore, α ∈ L which is absurd. Hence, R is j -Noetherian.

( 2 ) ⇒ ( 3 ) Let ( I n ) n ∈ N be an ascending chain of j -ideal of R . If I = ⋃ n ∈ N I n , then I is a j -ideal of R and, by the hypothesis, I is finitely generated. This makes I = R a 1 + … + R a p for some a 1 , … , a p ∈ I . Hence, there exists k ∈ N such that I ⊆ I k . Therefore, I n = I k for any n ≥ k and hence ( I n ) n ∈ N is stationary.

( 3 ) ⇒ ( 4 ) Let I be a j -ideal of R . Let L 1 ∕ I ⊆ L 2 ∕ I ⊆ … be an ascending chain of non-zero ideals of R ∕ I . As a result, L 1 ⊆ L 2 ⊆ … is an ascending chain of j -ideal of R and hence, by the hypothesis, there exists k ∈ N such that L n + 1 = L n for every n > k . Therefore, L n + 1 ∕ I = L n ∕ I for every n > k and hence ( L n ∕ I ) n ∈ N is stationary. Thus, R ∕ I is Noetherian.

( 4 ) ⇒ ( 1 ) Let Ω be a non-empty set of j -ideals of R which is not satisfying the property in (1). Then, for every I ∈ Ω , there exists J ∈ Ω such that I ⊆ J and J ⊈ I . Let I ∈ Ω and set Θ = { J ∈ Ω ∣ I ⊆ J } . Then, Θ does not have a maximal element. Hence, Λ = { J ∕ I ∣ J ∈ Θ } is a set of ideals of R ∕ I which also does not have a maximal element, which contradicts the fact that R ∕ I is Noetherian.□

Now, we are going to prove the Cohen-type theorem for j -Noetherian rings.

Theorem 2.2

A ring R is j -Noetherian if and only if its prime j -ideals are finitely generated.

Proof

If R is j -Noetherian, it is obvious that all prime j -ideals of R are finitely generated. Now, suppose that all prime j -ideals of R are finitely generated and assume that R is not j -Noetherian. Therefore, the set F of all j -ideals that are not finitely generated is a non-empty set that is ordered by the inclusion. Let ( I λ ) λ be a totally ordered family of elements of F . Then, I = ⋃ J λ is a j -ideal. If I = ( a 1 , … , a n ) for some a 1 , … , a n ∈ R , there exists λ such that I λ = ( a 1 , … , a n ) which is absurd. Hence, F admits a maximal element P . We will show that P is a prime ideal of R . This makes P a prime j -ideal that is finitely generated, which is a contradiction to the fact that P ∈ F . Suppose that there exists a , b ∈ R \ P such that a b ∈ P . Since P ⊊ P + a R , it follows that P + a R is j -ideal that is finitely generated. Set P + a R = ( α 1 + a x 1 , … , α n + a x n ) for some α i ∈ P and x i ∈ R . Consider the ideal J = ( P : a ) = { y ∈ R ; y a ∈ P } . Note that P ⊊ J as b ∈ J \ P . Therefore, J is a finitely generated j -ideal. Put J = ( β 1 , … , β s ) for some β 1 , … , β s ∈ R . We will show that P = ( α 1 , … , α n , a β 1 , … , a β s ) , which is the desired contradiction. Since β i ∈ J and a β i ∈ P , we obtain that ( α 1 , … , α n , a β 1 , … , a β s ) ⊆ P . Conversely, let x ∈ P ⊆ P + a R . There exist u 1 , … , u n ∈ R such that x = u 1 ( α 1 + a x 1 ) + … + u n ( α n + a x n ) = u 1 α 1 + … + u n α n + ( u 1 x 1 + … + u n x n ) a . It suffices to show that ( u 1 x 1 + … + u n x n ) a ∈ ( a β 1 , … , a β s ) . Note that ( u 1 x 1 + … + u n x n ) a = x − ( u 1 α 1 + … + u n α n ) ∈ P , and hence u 1 x 1 + … + u n x n ∈ J , by the definition of J . Since J = ( β 1 , … , β s ) , we obtain ( u 1 x 1 + … + u n x n ) a ∈ ( a β 1 , … , a β s ) .□

Remark 2.3

Note that every Noetherian ring is j -Noetherian, and if all prime ideals of R are j -ideals, then the two notions are identical. In particular, if Nil ( R ) is not prime, then the notions of Noetherian and nonnil-Noetherian ring are identical.

Let A be a ring and let E be an A -module. The set A ∝ E of all pairs ( r , m ) ∈ A × E with the componentwise addition and the multiplication defined by: ( r , m ) ( b , f ) = ( r b , r f + b m ) is a unitary commutative ring, called the trivial extension (or idealization) of A by E . Note that A ∝ E is a graded ring with ( A ∝ E ) 0 = A ∝ 0 , ( A ∝ E ) 1 = 0 ∝ E , and ( A ∝ E ) n = 0 for n ≥ 2 . Recall from [13, Theorem 3.3(1)] that an ideal J of A ∝ E is homogeneous if J = I ∝ N where I = { r ∈ A ∣ ( r , b ) ∈ J for some b ∈ E } and N = { m ∈ E ∣ ( s , m ) ∈ J for some s ∈ E } . The basic properties of the trivial ring extension are summarized in [10,13,14].

Example 2.4

Let A be a ring and let E be an A -module. If j is an ideal of the trivial ring extension R = A ∝ E , then the following three cases are possible:

If j is not a homogeneous ideal of R , then 0 ∝ E ⊈ j . Indeed, if 0 ∝ E ⊆ j , we can easily show that j = I ∝ E where I = { x ∈ A ∣ ( x , e ) ∈ j for some e ∈ R } is an ideal of A . Let e 0 ∈ E such that ( 0 , e 0 ) ∉ j . Since ( 0 , e 0 ) 2 = ( 0 , 0 ) , ( 0 , e 0 ) ∈ P for every prime ideal P of R , and so every prime ideal of R is a j -ideal. Consequently, according to Theorem 2.2, Cohen’s theorem, and [13, Theorem 4.8], R is a j -Noetherian ring if and only if A is a Noetherian ring and E is finitely generated.
If j is a homogeneous ideal of R of the form I ∝ F where I is an ideal of A and F ⊊ E is a sub-module of E , then 0 ∝ E ⊈ j . Indeed, if 0 ∝ E ⊆ j , we can easily show that F = E . Therefore, as in the first case, we obtain that R is a j -Noetherian ring if and only if A is Noetherian ring and E is finitely generated.
The case where j is a homogeneous ideal of R of the form I ∝ E , where I is an ideal of A , will be addressed in Corollary 2.22.

Let I be an ideal of a ring A . According to [15], the amalgamated duplication of A along I , denoted by A ⋈ I , is the sub-ring of A × A given by A ⋈ I ≔ { ( a , a + i ) ∣ a ∈ A and i ∈ I } . This extension has been examined, in a general context and from the perspective of pullbacks, by D’Anna and Fontana [16]. It is worth noting that the concept of amalgamated duplication of a ring R along an ideal I predates [15], originally termed as the double of R along I and denoted D ( R , I ) (see, for instance, [17]).

One of the main differences between the ring A ⋈ I and the ring A ∝ I is that the ring A ⋈ I can be reduced (in fact the ring A ∝ I is always reduced if A is an integral domain).

Example 2.5

Let I be an ideal of a ring A . If j is an ideal of R = A ⋈ I , then the following two cases are possible:

The first case is that 0 × I ⊈ j and I × 0 ⊈ j . Note that the prime ideals of R have the form P ′ = P ⋈ I ≔ { ( a , a + i ) ∣ a ∈ P and i ∈ I } , where P is a prime ideal of A or P ¯ = { ( a , a + i ) ∣ a ∈ A , i ∈ I , and a + i ∈ P } = { ( a + i , a ) ∣ a ∈ P and i ∈ I } , where P is a prime ideal of A not containing I . In particular, for every prime ideal Q of R , we have 0 × I ⊂ Q or I × 0 ⊂ Q . Thus, all prime ideals of R are j -ideals, and consequently R is a j -Noetherian ring if and only if R is a Noetherian ring if and only if A is a Noetherian ring (according to Theorem 2.2 and [16, Corollary 3.3]).
The second case is that 0 × I ⊂ j or I × 0 ⊂ j . If 0 × I ⊂ j , then j = K ⋈ I ≔ { ( a , a + i ) ∣ a ∈ K and i ∈ I } , where K = { a ∈ A ∣ ( a , a + i ) ∈ j for some i ∈ I } . However, if 0 × I ⊈ j , then I × 0 ⊂ j and we have j = { ( a , a + i ) ∣ a ∈ A , i ∈ I and a + i ∈ K } = { ( a + i , a ) ∣ a ∈ K and i ∈ I } , where K = { a ∈ A ∣ ( a + i , a ) ∈ j for some i ∈ I } is an ideal of A not containing I . This case will be discussed later in Corollary 2.21.

Note that if R is a Noetherian ring, then R is j -Noetherian for every ideal j of R , and the converse holds when j = 0 . Furthermore, if R is j -Noetherian, then R ∕ j is a Noetherian ring. The following example provides first, a class of ideals j for which the notions of Noetherian and j -Noetherian are the same, second, an example of a j -Noetherian ring that is not Noetherian, and third, an example of an ideal j of a ring R for which the ring R ∕ j is Noetherian while R is neither a Noetherian ring nor a j -Noetherian.

Example 2.6

Let R be a ring and j ⊊ Nil ( R ) an ideal of R . Then, R is a j -Noetherian ring if and only R is a Noetherian ring.
Let ( D , m ) be a local domain which is not a field, and let K be its quotient field. Set R = D ∝ K , then R is a ( m ∝ K ) -Noetherian ring which is not Noetherian by [13, Theorem 4.8] since K is not finitely generated.
Let ( D , m ) be a local domain which is not Noetherian where the maximal ideal m is a principal ideal, given as m = x D for some x ∈ R . If R = D ∝ M where M = D ∕ m , then R is not a Noetherian ring. If j = ( x , 1 ) R , then j is not a homogeneous ideal by [14, Example 2.5], and hence R is not a j -Noetherian ring according to Example 2.4. On the other hand, R ∕ j is a Noetherian ring. Indeed, if Q is a prime ideal of R ∕ j , then Q = ( P ∝ M ) + j for some prime ideal j ⊂ P ∝ M of R . Let d ∈ m where d = α x for some α ∈ D . Hence, ( d , α ¯ ) = ( α , 0 ) ( x , 1 ) ∈ j ⊂ P ∝ M , and so m = P . Consequently, Q is finitely generated, and hence R ∕ j is a Noetherian ring.

Let j be an ideal of a ring R . If R is a j -Noetherian ring, then it is clear that R is a j -Noetherian ring. However, the converse is not always true, as the following example shows.

Example 2.7

Let ( D , m ) be a local domain which is not Noetherian where the maximal ideal m is a principal ideal, given as m = x D for some x ∈ R . Let R = D ∝ M where M = D ∕ m . If j = ( x , 1 ) R , then R is not a j -Noetherian ring according to Example 2.6. However, j = m ∝ M (by the proof of [18, Example 2.32(3)]), and so ( R , j ) is a local ring. Therefore, R is naturally a j -Noetherian ring.

Remark 2.8

A nonnil-Noetherian ring which is not Noetherian can be seen as a j -Noetherian ring, which is not j -Noetherian, where j = ( 0 ) in this case.

Proposition 2.9

Let R be a ring and j a proper ideal of R. Then, R is j -Noetherian if and only if R satisfies the ascending chain condition for finitely generated j -ideals.

Proof

It suffices to show the implication ( ⇐ ) . Assume that there is a j -ideal I of R that is not finitely generated. Let a 1 ∈ I \ j , then ( a 1 ) ⊂ I . Let a 2 ∈ I \ ( a 1 ) , then ( a 1 ) ⊂ ( a 1 , a 2 ) ⊂ I , and so on. Since a 1 ∉ j , we have a strictly ascending chain of finitely generated j -ideals.□

Proposition 2.10

Let f : A ⟶ B be a surjective homomorphism of rings and j a proper ideal of A . If A is j -Noetherian, then B is f ( j ) -Noetherian.

Proof

Assume that A is j -Noetherian. Let I be an f ( j ) -ideal of B . There exists y = f ( x ) ∈ I such that y ∉ f ( j ) . Therefore, x ∈ f − 1 ( I ) and x ∉ j , and so f − 1 ( I ) is a j -ideal of A . Thus, f − 1 ( I ) is finitely generated of A , and hence I = f ( f − 1 ( I ) ) is a finitely generated ideal of B .□

Any ring admits trivial idempotent elements 0 and 1. Recall that a ring R is said to be decomposable if R admits a non-trivial idempotent. Let Idem ( R ) denote the set of idempotent elements of R .

Theorem 2.11

Let R be a decomposable ring and j an ideal of R. Assume that e j ≠ ( e ) for each e ∈ Idem ( R ) − { 0 , 1 } . Then, the following statements are equivalent.

R is a Noetherian ring.
R is a j -Noetherian ring.
For each e ∈ Idem ( R ) − { 0 , 1 } , R ∕ ( e ) is a Noetherian ring.
If e ∈ Idem ( R ) − { 0 , 1 } , then every ideal of R contained in ( e ) is finitely generated.

Proof

(1) ⇒ (2) Straightforward.

(2) ⇒ (3) Consider e ∈ Idem ( R ) − { 0 , 1 } . Let I be an ideal of R which contains ( e ) . Since e j ≠ ( e ) , e ∉ j , and then I ⊈ j . Hence, I is finitely generated. So, R ∕ ( e ) is Noetherian, as desired.

(3) ⇒ (4) Let e ∈ Idem ( R ) − { 0 , 1 } and consider an ideal I of R such that I ⊆ ( e ) . Clearly, I e = I . If I = ( 0 ) , the result is clear. So, we may assume that I ≠ 0 . In this case, we certainly have I ⊈ ( 1 − e ) . Since 1 − e ∈ Idem ( R ) − { 0 , 1 } , R ∕ ( 1 − e ) is a Noetherian ring. Thus, I + ( 1 − e ) is a finitely generated ideal of R . Consequently, I = I e = ( I + ( 1 − e ) ) e is a finitely generated ideal of R .

(4) ⇒ (1) Let I be an ideal of R . Since I e and I ( 1 − e ) are finitely generated, I = I e + I ( 1 − e ) is finitely generated. Hence, R is Noetherian.□

Remark 2.12

Let R be a decomposable ring and j an ideal of R . Suppose that R = R 1 × R 2 where R 1 and R 2 are two rings and j = j 1 × j 2 where j 1 = R 1 or j 2 = R 2 . Without loss of generality, assume that j 1 = R 1 , and consequently j 2 ≠ R 2 . It is easy to verify that an ideal I = I 1 × I 2 of R is a j -ideal if and only if I 2 is a j 2 -ideal. Since I = 0 × I 2 is a finitely generated ideal of R for every j 2 -ideal I 2 of R 2 , we obtain that R 2 is j 2 -Noetherian. On the other hand, for every ideal I 1 of R , we have I = I 1 × R 2 is a j -ideal of R , and so it is finitely generated. Hence, R 1 is a Noetherian ring. Conversely, it is easy to verify that R is a j -Neotherian ring if R 1 is a Noetherian ring and R 2 is a j 2 -Noetherian ring. Therefore, we conclude that R is a j -Neotherian ring if and only if R 1 is a Noetherian ring and R 2 is a j 2 -Noetherian ring.

Corollary 2.13

Let n ≥ 2 be an integer, R 1 , … , R n commutative rings with identity, and let j 1 , … , j n be ideals of R 1 , … , R n , respectively, with j i ≠ R i for all i = 1 , … , n . Set j = ∏ i = 1 n j i . Then, the following assertions are equivalent.

∏ i = 1 n R i is a Noetherian ring.
∏ i = 1 n R i is a j -Noetherian ring.
For all i = 1 , … , n , R i is a Noetherian ring.

Remark 2.14

In Corollary 2.13, if there exists an index k ∈ { 1 , … , n } such that R k is a j k -Noetherian ring which is not a Noetherian ring, then ∏ i = 1 n R i is never a j -Noetherian ring. For example, if R 1 is not Noetherian, there exists an ideal I 1 ⊆ j 1 of R 1 which is not finitely generated, and so I 1 × R 2 × … × R n is a j -ideal of ∏ i = 1 n R i which is not finitely generated. Thus, ∏ i = 1 n R i is not a j -Noetherian ring.
Let R 1 and R 2 be two rings, and let j 1 and j 2 be ideals of R 1 and R 2 , respectively, such that j 1 = R 1 or j 2 = R 2 . Without loss of generality, assume that j 2 = R 2 . If R = R 1 × R 2 and j = j 1 × j 2 , then R is a j -Noetherian ring if and only if R 1 is a j 2 -Noetherian ring and R 2 is a Noetherian ring. This justifies why we assume that all j i ≠ R i in the previous corollary.
Note that Z ∕ 4 Z is a Noetherian ring. Let R = ∏ n = 1 ∞ Z ∕ 4 Z , j = ∏ n = 1 ∞ 2 Z ∕ 4 Z , and I = ⊕ n = 1 ∞ Z ∕ 4 Z . Then, I is a j -ideal of R which is not finitely generated. Thus, R is not a j -Noetherian ring. This shows that Corollary 2.13 is not generally extended to the case of an infinite product of ( j -)Noetherian rings.

Proposition 2.15

Let R be a ring, ( j i ) i ∈ λ a family of ideals of R, and j = ∩ i ∈ λ j i . Then, the following assertions are equivalent.

R is a j -Noetherian ring.
For all i ∈ λ , R is a j i -Noetherian ring.

Proof

(1) ⇒ (2) Follows from the fact that j ⊆ j i for each i ∈ λ .

(2) ⇒ (1) Let J be a j -ideal of R . There exists x ∈ J such that x ∉ j . Since j = ∩ i ∈ λ j i , x ∉ j i 0 for some i 0 ∈ λ . By the assumption, J is a finitely generated ideal of R , since R is j i 0 -Noetherian. Thus, R is a j -Noetherian ring.□

Corollary 2.16

Let R be a ring. Then, the following assertions are equivalent.

R is a nonnil-Noetherian ring.
For every prime ideal P of R, R is a P-Noetherian ring.

Remark 2.17

Clearly, if R is a nonnil-Noetherian ring, then for every maximal ideal m of R , R is a m -Noetherian ring (by Corollary 2.16). However, the converse is not true in general. To see this, let ( R , j ) be a local domain which is not Noetherian. Since the only j -ideal of R is R itself, we obtain that R is a j -Noetherian ring. However, R is not a nonnil-Noetherian ring.

Let P ⊊ Q be two prime ideals of a ring R . The ideal Q is said to be of height 1 over P if there is no prime ideal P ′ satisfying that P ⊊ P ′ ⊊ Q .

Theorem 2.18

If j is a prime ideal of a ring R, then the following assertions are equivalent.

R is a j -Noetherian ring.
R ∕ j is a Noetherian ring, and every prime ideal of R that is of height 1 over j is finitely generated.

Proof

(1) ⇒ (2) A nonzero prime ideal of R ∕ j is of the form P ∕ j with P ∈ Spec ( R ) and j ⊊ P . By the hypothesis, P is finitely generated, which makes P ∕ j finitely generated. By Cohen’s theorem, R ∕ j is Noetherian. Let P be a prime ideal of R of height 1 over j . Note that P is a j -ideal as j ⊂ P . Thus, P is finitely generated.

(2) ⇒ (1) Let P be a prime j -ideal of R . Therefore, j ⊂ P and P ¯ = P ∕ j is a nonzero prime ideal of the integral domain R ∕ j . Let 0 ¯ ≠ x ¯ ∈ P ¯ and let Q ¯ ⊆ P ¯ be a prime ideal of R ∕ j that is minimal over ( x ¯ ) . Since R ∕ j is Noetherian, by Krull’s Principal Ideal Theorem, Q ¯ is of height 1. On the other hand, note that Q ¯ = Q ∕ j with j ⊈ Q ⊆ P , and Q is a height 1 over j . By the hypothesis, Q is finitely generated. By the isomorphism, we have ( ( R ∕ j ) ∕ ( Q ∕ j ) ) ≃ R ∕ Q , and hence the domain R ∕ Q is Noetherian. Note that P ∕ Q is finitely generated. Hence, P is finitely generated, and R is j -Noetherian by Theorem 2.2.□

A prime ideal P of a ring R is called divided if P ⊆ x R for every x ∈ R \ P .

Theorem 2.19

If j is a divided prime ideal of a ring R, then R is j -Noetherian if and only if R ∕ j is Noetherian domain.

Proof

By Theorem 2.18, it suffices to prove that if R ∕ j is Noetherian, then every prime ideal P of R that is of height 1 over j is finitely generated. Since R ∕ j is Noetherian, the ideal P ∕ j is finitely generated. Set P ∕ j = ( x ¯ 1 , … , x ¯ n ) with x 1 , … , x n ∈ P . If all the x i ∈ j , then P ∕ j = ( 0 ¯ ) and hence P = j , which is impossible since P is of height 1 over j . Therefore, there is x i 0 ∉ j . Since j is divided, it follows that j ⊂ ( x i 0 ) . Thus, P = j + ( x 1 , … , x n ) ⊆ ( x i 0 ) + ( x 1 , … , x n ) = ( x 1 , … , x n ) ⊆ P . Accordingly, P = ( x 1 , … , x n ) and hence P is finitely generated.□

In particular if j = Nil ( R ) is a divided prime ideal of R , we conclude easily the following corollary from [1, Theorem 1.2] and Theorem 2.18.

Corollary 2.20

Let R be a ring. If Nil ( R ) is a divided prime ideal, then R is a nonnil-Noetherian ring if and only if R ∕ Nil ( R ) is a Noetherian domain.

Let I be a nonzero ideal of a ring A . If J is a divided prime ideal of the ring R = A ⋈ I , then J has the form P ′ = P ⋈ I , where P is a divided prime ideal of A such that I ⊂ P and I = a I for every a ∈ R \ P (by [19, Lemma 3.7]).

Corollary 2.21

Let A be a ring, I be a nonzero ideal of A, and P be a divided prime ideal of A such that I ⊂ P and I = a I for every a ∈ R \ P . Consider R = A ⋈ I and j = P ⋈ I . Then, R is a j -Noetherian ring if and only if A is a P-Noetherian ring.

Let A be a ring, E an A -module, and consider the trivial ring extension R = A ∝ E . If P is a prime ideal of A then, following [19, Lemma 3.1], P ∝ E is a divided prime ideal of R if and only if P is a divided prime ideal of A and E = a E for each a ∈ A \ P .

Corollary 2.22

Let A be a ring, E an A-module, and P a divided prime ideal of A such that E = a E for each a ∈ A \ P . If R = A ∝ E is the trivial ring extension, then R is a ( P ∝ E ) -Noetherian ring if and only if A is a P-Noetherian ring.

Let P be a divided prime ideal of a ring R . The map ϕ P : R → K ≔ R P given by ϕ P ( a ) = a ∕ 1 for every a ∈ R is a ring homomorphism from R into K . Note that if P is a divided prime ideal of R , then ϕ P ( P ) is a divided prime ideal of ϕ P ( R ) .

Lemma 2.23

If P is a divided prime ideal of a ring R, then R ∕ P is ring-isomorphic to ϕ P ( R ) ∕ ϕ P ( P ) .

Proof

Define the map α : R ⟶ ϕ P ( R ) ∕ ϕ P ( P ) as α ( a ) = ϕ P ( a ) + ϕ P ( P ) for every a ∈ R . It is clear that α is a ring-homomorphism from R onto ϕ P ( R ) ∕ ϕ P ( P ) . We claim that Ker ( α ) = P . Indeed, it is easy to see that P ⊆ Ker ( α ) . Conversely, if a ∈ Ker ( α ) , there exists x ∈ P such that ϕ P ( a ) = ϕ P ( x ) , and consequently there exists t ∈ R \ P such that t ( a − x ) = 0 . Since P is a prime ideal of R , we obtain a − x ∈ P and hence a ∈ P . Therefore, R ∕ P is ring-isomorphic to ϕ P ( R ) ∕ ϕ P ( P ) .□

Corollary 2.24

If P is a divided prime ideal of a ring R, then the following statements are equivalent.

R is a P-Noetherian ring.
R ∕ P is a Noetherian domain.
ϕ P ( R ) ∕ ϕ P ( P ) is a Noetherian domain.
ϕ P ( R ) is a ϕ P ( P ) -Noetherian ring.

Proof

( 1 ) ⇒ ( 2 ) Follows from Theorem 2.19.

( 2 ) ⇒ ( 3 ) By Lemma 2.

( 3 ) ⇒ ( 4 ) Since ϕ P ( P ) is a divided prime ideal of ϕ P ( R ) , the claim is clear by Theorem 2.19.

( 4 ) ⇒ ( 1 ) Assume that ϕ P ( R ) is a ϕ P ( P ) -Noetherian ring. Since ϕ P ( P ) is a divided prime ideal of ϕ P ( R ) , ϕ P ( R ) ∕ ϕ P ( P ) is a Noetherian domain by Theorem 2.19. Hence, R ∕ P is a Noetherian domain by Lemma 2. Thus, R is a P -Noetherian ring by Theorem 2.19.□

Corollary 2.25

Let R be a j -Noetherian ring and let j ⊆ P ⊂ Q be two prime ideals of R. If there exists a prime ideal of R strictly between P and Q, then there exists an infinity of prime ideals between P and Q.

Proof

By Theorem 2.19, P ∕ j ⊂ Q ∕ j are two prime ideals of the Noetherian ring R ∕ j . If L is a prime ideal of R such that P ⊂ L ⊂ Q , then L ∕ j is a prime ideal of R ∕ j such that P ∕ j ⊂ L ∕ j ⊂ Q ∕ j . As a result, there exists an infinity of prime ideals of R ∕ j between the ideals P ∕ j and Q ∕ j . Therefore, there exists an infinity of prime ideals between P and Q .□

Let R be a commutative ring with identity, K the total quotient ring of R , j an ideal of R , and φ a generalized multiplicative system of R , i.e., φ is a multiplicative set of ideals of R . The φ -transform of R (or the generalized transform of R with respect to φ ) is an overring R φ ≔ { x ∈ K ∣ x A ⊆ R for some A ∈ φ } of R and j φ ≔ { x ∈ K ∣ x A ⊆ j for some A ∈ φ } is an ideal of R φ containing j .

Proposition 2.26

Let R be a commutative ring and T = R φ be a flat overring of R. If R is a j -Noetherian ring, then T is also a j φ -Noetherian ring.

Proof

Let T be a flat overring of R . Then, there exists a generalized multiplicative system φ of R such that T = R φ and A T = T for all A ∈ φ . Let Q be a prime j φ -ideal of T and let P = Q ∩ R . Note that P is a prime j -ideal of R . Indeed, if P ⊆ j , then for any x ∈ Q there exists A ∈ φ such that x A ⊆ P , and so x A ⊆ P . This makes x ∈ j φ and hence Q ⊆ j φ , which is impossible. Since R is a j -Noetherian ring, there exist c 1 , … , c n ∈ R such that c 1 R + … + c n R = P . Let x ∈ Q . Since Q = P φ , there exists an element A ∈ φ such that x A ⊆ P ; so we obtain x ∈ P T ⊆ c 1 T + … + c n T . Hence, c 1 T + … + c n T = Q , which implies that Q is finitely generated. Thus, by Theorem 2.2, T is a j φ -Noetherian ring.□

Corollary 2.27

Let R be a commutative ring, j an ideal of R, and S a multiplicative subset of R. If R is a j -Noetherian ring, then R S is a j S -Noetherian ring.

Proposition 2.28

Let R ⊆ T be an extension of rings such that J T ∩ R = J for each ideal J of R, and let j be an ideal of R. If T is a j T -Noetherian ring, then R is a j -Noetherian ring.

Proof

Let J be a j -ideal of R . Since J T ∩ R = J , it is easy to verify that J T is a j T -ideal of T . Hence, we can find a 1 , … , a n ∈ j T such that ( a 1 , … , a n ) = j T since T is a j T -Noetherian ring. Let F be a finitely generated sub-ideal of j such that ( a 1 , … , a n ) ⊆ F T . Then, we have I = j T ∩ R ⊆ F T ∩ R = F . Thus, R is a j -Noetherian ring.□

Proposition 2.29

If R is a j -Noetherian ring, then each j -ideal contains a power of its radical.

Proof

Let I be a j -ideal of the j -Noetherian ring R . Then, I ⊆ I and hence I is a finitely generated ideal. Let I = x 1 R + … + x s R . There exists an element n ∈ N * , such that x i n ∈ I for any 1 ≤ i ≤ s . Let m = s n . Then, ( I ) m is generated by the products of the form x 1 r 1 … x s r s , with r 1 + … + r s = m . Hence, there exists 1 ≤ i ≤ s , such that r i ≥ n . Therefore, x 1 r 1 … x s r s ∈ I and ( I ) m ⊆ I .□

The following theorem investigates the behavior of polynomial and power series ring extensions for j -Noetherian rings. For a ring R , let Z ( R ) denote the set of zero-divisors of R .

Theorem 2.30

The following assertions are equivalent for a ring R .

The ring R is Noetherian.
The ring R [ X ] is j -Noetherian for every ideal j ⊂ Z ( R [ X ] ) .
The ring R [ X ] is j 0 -Noetherian for some ideal j 0 ⊂ Z ( R [ X ] ) .
The ring R [ [ X ] ] is j -Noetherian for every ideal j ⊂ Z ( R [ [ X ] ] ) .
The ring R [ [ X ] ] is j 0 -Noetherian for some ideal j 0 ⊂ Z ( R [ [ X ] ] ) .

Proof

( 1 ) ⇒ ( 2 ) ⇒ ( 3 ) and ( 1 ) ⇒ ( 4 ) ⇒ ( 5 ) Straightforward.

( 3 ) ⇒ ( 1 ) Assume that R [ X ] is j 0 -Noetherian for some ideal j 0 ⊂ Z ( R [ X ] ) . Let P be a prime ideal of R . Since X ∉ Z ( R [ X ] ) , P + X R [ X ] is a j 0 -ideal of R and it is finitely generated. Hence, there exist P 1 , … , P n ∈ R [ X ] such that

P 1 R [ X ] + … + P n R [ X ] = P + X R [ X ] .

As a result, we obtain

P 1 ( 0 ) R + … + P n ( 0 ) R = P .

Thus, P is finitely generated ideal of R . Then, by Cohen’s theorem, R is Noetherian.

( 5 ) ⇒ ( 1 ) Similar to ( 3 ) ⇒ ( 1 ) .□

Since Nil ( R [ X ] ) ⊆ Z ( R [ X ] ) and Nil ( R [ [ X ] ] ) ⊆ Z ( R [ [ X ] ] ) , we can use the previous theorem to deduce [3, Theorem 3.3].

Corollary 2.31

The following assertions are equivalent for a ring R .

R is a Noetherian ring.
R [ X ] is a nonnil-Noetherian ring.
R [ [ X ] ] is nonnil-Noetherian ring.

Next we study some properties of polynomial and power series rings over j -Noetherian rings.

Theorem 2.32

Let R be a j -Noetherian ring with j finitely generated and P a prime ideal of R [ X ] such that j R [ X ] ⊊ P .Then, P is a finitely generated ideal of R [ X ] .

Proof

Let P be a prime ideal of R [ X ] such that j ( R [ X ] ) ⊊ P . Let A be the subset of P which consists of polynomials with a leading coefficient not in j . Then, A ≠ ∅ and P = A + j R [ X ] . By the assumption, j is a finitely generated ideal of R . Note that j R [ X ] is a finitely generated ideal of R [ X ] , and hence we can find a 1 , … , a q ∈ R such that

j R [ X ] = ( a 1 , … , a q ) .

For each integer n ≥ 0 , let P n be the ideal of R generated by the leading coefficients of the polynomials of degree less than or equal to n in A and P n = 0 if there is no polynomial of this form. Let r be the smallest non-negative integer such that P r ≠ ( 0 ) . Then, P r ⊆ P r + 1 ⊆ P r + 2 ⊆ … is an ascending chain of j -ideals of R . Set J = ⋃ n ≥ r P n . Then, J is a j -ideal of R . Since R is a j -Noetherian ring, there exist b 1 , … , b m ∈ R such that

j = ( b 1 , … , b m ) .

For each i = 1 , … , m , take an element f i ∈ ( A ) with leading coefficient b i and let d i be the degree of f i . Set d = max { d i ∣ i = 1 , … , m } . Since P k is a j -ideal of R for all k ∈ { r , … , d } , P k is a finitely generated ideal of R and hence there exist a k 1 , … , a k n k ∈ R such that

P k = ( a k 1 , … , a k n k ) .

For each k ∈ { r , r + 1 , … } and i ∈ { 1 , … , n k } , we can take an element f k i ∈ A with leading coefficient a k i and degree k . Let f = ∑ i = 0 v p i X i ∈ P . If v > d , then p v ∈ ( b 1 , … , b m , a 1 , … , a q ) and hence we obtain

p v = r 1 b 1 + … + r m b m + r m + 1 a 1 + … + r m + q a q .

for some r 1 , … , r m + q ∈ R . Therefore,

f − ( r 1 f 1 X v − d 1 + … + r m f m X v − d m + r m + 1 a 1 X v + … + r m + q a q X v )

is a polynomial in P with degree less than or equal to v − 1 . By repeating this process, we can obtain a polynomial h ≔ ∑ i = 0 ℓ q i X i ∈ P with degree less than or equal to d . Now, q ℓ ∈ ( a ℓ 1 , … , a ℓ n ℓ ′ a 1 , … , a q ) and hence we obtain

q ℓ = c 1 a ℓ 1 + … + c n ℓ a ℓ n ℓ + c n ℓ + 1 a 1 + … + c n ℓ + q a q

for some c 1 , … , c n ℓ + q ∈ R . Consequently,

h − ( c 1 f ℓ 1 + … + c n ℓ f ℓ n ℓ + c n ℓ + 1 a 1 X ℓ + … + c n ℓ + q a q X ℓ )

is a polynomial with degree less than or equal to ℓ − 1 . By repeating this process, we obtain

∏ k = r d t k f = g 1 + g 2

for some g 1 ∈ ( f 1 , … , f m , f r 1 , … , f d n d , a 1 , … , a q ) and g 2 ∈ P with deg ( g 2 ) ≤ r − 1 . If g 2 = 0 , then ∏ k = r d t k f ∈ ( f 1 , … , f m , f r 1 , … , f d n d , a 1 , … , a q ) . Suppose that g 2 ≠ 0 and write g 2 = ∑ i = 0 r − 1 e i X i . By the construction of A , we obtain e r − 1 ∈ I ; so e r − 1 = a 1 z 1 + … + a q z q for some z 1 , … , z q ∈ R . Therefore, g ≔ g 2 − ( a 1 z 1 X r − 1 + … + a q z q X r − 1 ) ∈ P with degree less than or equal to r − 2 . By the construction of A , the leading coefficient of g belongs to j . By repeating this process, we obtain:

g 2 ∈ ( a 1 , … , a q ) .

Hence, we obtain

∏ k = r d t k f ∈ ( f 1 , … , f m , f r 1 , … , f d n d , a 1 , … , a q ) .

Since f is arbitrarily chosen, we obtain

P = ( f 1 , … , f m , f r 1 , … , f d n d , a 1 , … , a q ) ,

which means that P is a finitely generated ideal of R [ X ] .□

Proposition 2.33

Let R be a commutative ring. If R is a j -Noetherian ring, then any ideal of R [ [ X ] ] properly containing j + X R [ [ X ] ] is a finitely generated ideal of R [ [ X ] ] .

Proof

Let A be an ideal of R [ [ X ] ] properly containing j + X R [ [ X ] ] . Then, A contains X and so A = ( A ∩ R ) + X R [ [ X ] ] . Therefore, A ∩ R is a j -ideal of R . Since R is a j -Noetherian ring, A ∩ R is a finitely generated ideal of R , and hence there exists r 1 , … , r n ∈ R such that ( A ∩ R ) = ( r 1 , … , r n ) . Consequently, A = ( r 1 , … , r n , X ) . Thus, A is a finitely generated ideal of R [ [ X ] ] .□

Proposition 2.34

Let R be a j -Noetherian ring, P ∈ spec ( R [ [ X ] ] ) , and set p = { f ( 0 ) ; f ∈ P } . If p ⊈ j , then P is finitely generated.

Proof

As p is an ideal of R which is not contained in j , p is finitely generated. Set p = ( a 1 , … , a n ) for some a 1 , … , a n ∈ p . For any 1 ≤ i ≤ n , let f i ∈ P such that f i ( 0 ) = a i . Then, two cases are possible.

Case 1. If X ∈ P : We show that P = ( f 1 , … , f n , X ) . In fact, ( f 1 , … , f n , X ) ⊆ P is clear. Conversely, let f ∈ P . Thus, f ( 0 ) ∈ p and hence there exist α 1 , … , α n in R such that f ( 0 ) = ∑ i = 1 n α i a i . Therefore, f − ∑ i = 1 n α i f i ∈ X R [ [ X ] ] which makes f ∈ ( f 1 , … , f n , X ) .

Case 2. If X ∉ P : We show that P = ( f 1 , … , f n ) . In fact, ( f 1 , … , f n ) ⊆ P is clear. Conversely, if f ∈ P , then f ( 0 ) ∈ p and hence f ( 0 ) = ∑ i = 1 n r 0 i a i for some r 01 , … , r 0 n ∈ R . Therefore, f − ∑ i = 1 n r 0 i f i = X f 1 for some f 1 ∈ R [ [ X ] ] . Since X f 1 ∈ P and X ∉ P , it follows that f 1 ∈ P and hence f 1 − ∑ i = 1 n r 1 i f i = X f 2 for some r 11 , … , r 1 n ∈ R and f 2 ∈ P . Now, by continuing this process, we obtain

f = ∑ i = 1 n f i ∑ j = 0 ∞ r j i X j .

Thus, in both cases we have that P is a finitely generated ideal of R [ [ X ] ] .□

Acknowledgments

The authors thank the anonymous referee for the careful review and the valuable comments which improved the presentation of the article.

Funding information: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small group Research Project under grant number RGP1/276/44.
Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-12-21

Revised: 2024-04-04

Accepted: 2024-04-11

Published Online: 2024-05-17

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

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https://doi.org/10.1515/math-2024-0014

Keywords for this article

𝒿-Noetherian rings; 𝒿-ideals; nonnil-Noetherian rings

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