On nonnil-coherent modules and nonnil-Noetherian modules

Younes El Haddaoui; Hwankoo Kim; Najib Mahdou

doi:10.1515/math-2022-0526

Open Access Published by De Gruyter Open Access November 24, 2022

On nonnil-coherent modules and nonnil-Noetherian modules

Younes El Haddaoui , Hwankoo Kim and Najib Mahdou

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0526

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:

R is a nonnil-coherent ring.
R is a nonnil-coherent R-module.
Every finitely generated free R-module is nonnil-coherent.
Every finitely presented module is nonnil-coherent.
Every finitely generated ϕ -submodule of a finitely presented R-module is finitely presented.
Any direct product of ϕ -flat R-modules is ϕ -flat.
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:

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

Note that for a ϕ -torsion R -module M , we have
M is nonnil-Noetherian ⇔ M is Noetherian .
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:

R is nonnil-coherent.
R s is a finitely presented ideal of R for any s ∈ R ⧹ Nil ( R ) .
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:

R is a nonnil-coherent ring.
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 ) .
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

Z ∝ Q is a nonnil-coherent ring.
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:

R = A ⋈ f J ∈ ℋ .
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:

A ⋈ f J is a nonnil-Noetherian ring.
A ⋈ f ¯ J N ( J ) is a Noetherian domain.
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:

A ⋈ f J is a ϕ -coherent ring.
A ⋈ f ¯ J N ( J ) is a coherent domain.
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:

A ⋈ f J is a nonnil-coherent ring,
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:

A ⋈ f J is a nonnil-coherent ring.
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.

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

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