RINGS WITH DIVISIBILITY ON ASCENDING CHAINS OF IDEALS

. According to Dastanpour and Ghorbani, a ring R is said to satisfy divisibility on ascending chains of right ideals ( ACC d ) if, for every ascending chain of right ideals I 1 ⊆ I 2 ⊆ I 3 ⊆ I 4 ⊆ ... of R , there exists an integer k ∈ N such that for each i ≥ k , there exists an element a i ∈ R such that I i = a i I i +1 . In this paper, we examine the transfer of the ACC d -condition on ideals to trivial ring extensions. Moreover, we investigate the connection between the ACC d on ideals and other ascending chain conditions. For example we will prove that if R is a ring with ACC d on ideals, then R has ACC on prime ideals.


Introduction
In [5], extending the notion of ACC on right ideals (i.e. right noetherian rings), a ring R is said to satisfy divisibility on ascending chains of right ideals (ACC d on right ideals, for short) if, for every ascending chain of right ideals I 1 ⊆ I 2 ⊆ I 3 ⊆ I 4 ⊆ . . . of R, there exists an integer k ∈ N such that for each i ≥ k, there exists an element a i ∈ R such that I i = a i I i+1 . If R is commutative and all the multiple factors a i are invertible, then R is noetherian.
In [5], Dastanpour and Ghorbani investigated thoroughly the notion of ACC d on right ideals, highlighting some of its properties and obtaining several interesting results in the commutative case. For example, they prove that every commutative semilocal ring that satisfies ACC d on ideals has a finitely generated socle and has only finitely many minimal prime ideals.
In this paper we focus our attention on commutative rings and continue the investigation that was carried out by Dastanpour and Ghorbani in [5]. In particular, we provide sufficient conditions for the trivial extension of rings to satisfy the ACC d on ideals. Moreover, we consider the connection between ACC d and other ascending chain conditions on ideals. For example, we prove that if R satisfies ACC d on ideals, then R satisfies ACC on prime ideals.
Throughout this paper, all rings are commutative with identity, and all modules are unital. If R is a ring, we denote by N il(R) to the set (ideal) of all nilpotent elements of R. When A is a local ring with M as its unique maximal ideal, we will write and say (A, M ) is local.

Main results
Let A be a ring and E an A-module. Then A ⋉ E, the trivial (ring) extension of A by E, is the ring whose additive structure is that of the external direct sum A⊕E and whose multiplication is defined by (a, e)(b, f ) := (ab, af +be) for all a, b ∈ A and all e, f ∈ E. The basic properties of trivial ring extensions are summarized in [6] and [7]. Moreover, interesting examples and constructions of trivial ring extensions could be found in [1], [3] and [8].
In [5,Proposition 2.3], the authors proved that homomorphic images of rings with ACC d on right ideals satisfy the ACC d on right ideals. We will use this result to establish our next theorem on the transfer of the ACC d -condition on ideals to trivial ring extensions.
Theorem 2.1. Let A be a ring, E a nonzero A-module, and R := A ∝ E the trivial ring extension of A by E. Then: (1) If R satisfies ACC d , then so is A.  (2a). Assume that R satisfies ACC d . We need to show that A is noetherian. Let . be an ascending chain of ideals of A. Since I 1 ∝ E ⊆ . is an ascending chain of ideals of the ring R, there exists an integer k ∈ N such that for each i ≥ k, there is an element We claim that a i / ∈ M . Otherwise, assume that a i ∈ M . In this case, since This means that f i = 0 for all f i ∈ E, a clear contradiction since E ̸ = 0. Now, inasmuch as (A, M ) is a local ring and a i / ∈ M , we infer that a i is invertible, and  Remark 2.3. Theorem 2.1 above shows that if A is a ring, E is a nonzero Amodule, and R := A ∝ E has the ACC d , then so is A. However, Example 2.2 shows that the converse need not be true. Moreover, we will construct below an example to show that if E is not finitely generated, then statement (2b) of Theorem 2.1 need not be true. But first, the next result will be needed to construct the example. (2) If P is a prime ideal of A such that P 2 = 0, then, A P satisfies ACC d if and only if A P is noetherian, where A P is the localization of A with respect to the prime ideal P .

Let E be an
Proof. (1) We only need to establish the forward implication. To see this, assume that A satisfies ACC d and let I 1 ⊆ I 2 ⊆ I 3 ⊆ I 4 ⊆ . . . be an ascending chain of nonzero ideals of A. We will show that the chain is stationary. By the ACC d , there as required.
(2) If P is a prime ideal of A with P 2 = 0, then (A P , (P A P )) is a local ring with (P A P ) 2 = 0. Now an application of part (1)  In the next theorem we establish one of the main results of this paper. More precisely, we will show that if A is a ring satisfying the ACC d -condition, then A satisfies the ACC on prime ideals. But first, we need to prove a couple of lemmas. Proof. Observe first that x / ∈ n i=1 P i . Otherwise, the equation n i=1 P i = xI implies the existence of a non-zero element y of I such that x = xy, a contradiction since A is an integral domain and I ̸ = A. With this observation in mind, we consider two cases: Case 1: Suppose that x / ∈ n i=1 P i . Since x / ∈ P i , xI ⊆ P i and P i is a prime ideal, 1 ≤ i ≤ n, we infer that I ⊆ P i , 1 ≤ i ≤ n. Therefore I ⊆ n i=1 P i , and so I = n i=1 P i . Case 2: Suppose that x ∈ n i=1 P i , and let J be a subset of {1, 2, . . . .., n} such that x ∈ i∈J P i and x / ∈ ( i / ∈J P i ). Now for each i / ∈ J, since P i is a prime ideal, xI ⊆ P i , and x / ∈ P i , we infer that I ⊆ P i for each i / ∈ J. Therefore xI ⊆ xP i for each i / ∈ J, and so (xI) ( i / ∈J xP i ) = xI. Moreover, since x ∈ P j , for each j ∈ J, it follows that xP i P j = xP i for each j ∈ J. Furthermore, we have for each j ∈ J, it follows that i / ∈J xP i = xI, and so x( i / ∈J P i ) = xI. Inasmuch as A is an integral domain, we infer that i / ∈J P i = I, as required. □ Lemma 2.8. If A is an integral domain satisfying the ACC d -condition, then the following hold: (1) A satisfies the ACC on ideals each of which is an intersection of a finite number of prime ideals. In particular, A satisfies the ACC on prime ideals.
(2) If whenever I and J are ideals of A, there exists a set of prime ideals . be an ascending chain of non-zero ideals of A such that, for each k ∈ N, I k = i∈J k P i , an intersection of a finite number of prime ideals P i . Since A satisfies the ACC d -condition, there exists an integer k such that for each n > k, there exists an element a n of A satisfying the relation I k = a n I n . Thus, i∈J k P i = a n ( i∈Jn P i ). By Lemma 2.7 above, there exists a subset S k ⊆ J k such that i∈S k P i = i∈Jn P i = I n . Inasmuch as the sequence {|S k |} is bounded and decreasing, it is convergent, where |S| denotes the number of elements of the set S. Therefore, the sequence {|S k |} is stationary. Let K ∈ N such that for each i ≥ K, S i = S K . Since I K ⊆ I i , it follows that I K = I i for each i ≥ K. This shows that the chain I 1 ⊆ I 2 ⊆ I 3 ⊆ I 4 ⊆ . . . is stationary, as required.
(2) This claim follows easily from (1). □ Theorem 2.9. Let A be a ring. If A satisfies ACC d , then A satisfies ACC on prime ideals.
Proof. We consider two cases: Case 1: A is an integral domain. In this case, apply Lemma 2.8.

Case 2:
A is not an integral domain. We need to show that A satisfies the ACC on prime ideals. To see this, let P 1 ⊆ P 2 ⊆ P 3 ⊆ P 4 ⊆ P 5 ⊆ . . . be an ascending chain of proper prime ideals of A. By [5, Proposition 2.3], since P 1 is a proper prime ideal of A, A/P 1 is an integral domain with the ACC d -condition. Now, by Lemma 2.8, the ascending chain P 2 /P 1 ⊆ P 3 /P 1 ⊆ P 4 /P 1 ⊆ P 5 /P 1 ⊆ . . . is stationary, and so is the chain P 1 ⊆ P 2 ⊆ P 3 ⊆ P 4 ⊆ P 5 ⊆ . . ., as required. □ The following example shows that the converse to Theorem 2.9 need not be true.
Example 2.10. Let K be a field, E an infinite dimensional K-vector space, and Then R satisfies the ACC on prime ideals but R does not satisfy the ACC d -condition.
In the next result we highlight some of the interesting features of the rings that satisfy the ACC d -condition. But first, we need the following lemma.
Lemma 2.11. Let A be a ring satisfying the ACC d -condition and I 1 ⊆ I 2 ⊆ I 3 ⊆ I 4 ⊆ . . . be a non-stationary ascending chain of ideals of A. Then for each n ≥ 1, I n is strictly contained in a proper principal ideal of A.
Proof. Since A satisfies the ACC d -condition, there exists an integer k such that Let n be a nonzero integer, and consider the following two cases: Case 1: If n ≥ k, then I n = x n I n+1 ⊆ x n A. As the chain is non-stationary, x n is not invertible, and so I n is properly contained in the principal ideal x n A. (1) Either I is strictly contained in a proper principal ideal of A or A/I is noetherian.
(2) If N il(A) is finitely generated that is not strictly contained in any proper principle ideal of A, then A is noetherian. Proof.
(1) If every ascending chain of ideals containing I is stationary, then A/I is noetherian. Otherwise, there exists a non-stationary ascending chain of ideals of A containing I. By Lemma 2.11, I is strictly contained in a proper principal ideal of A.
(2) By (1), A/N il(A) is noetherian, and so every ideal of A/N il(A) is finitely generated. In particular, P/N il(A) is finitely generated, where P is a prime ideal of A. Inasmuch as N il(A) is finitely generated, we infer that P is finitely generated.
This means that all prime ideals of A are finitely generated, and so A is noetherian. Next, we provide an example of a ring R with ACC on prime ideals that is not coherent.
Example 2.17. Let K be a field, E an infinite dimensional K-vector space, and R = K ∝ E. By Example 2.10, R satisfies the ACC on prime ideals. However, it is not difficult to see that R is not coherent.