About Weakly Bézout Rings

In this paper, we examine the transfer of the proprety weakly Bézout to the trivial ring extensions. These results provide examples of weakly Bézout rings that are not Bézout rings. We show that the proprety weakly Bézout is not stable under finite direct products. Also, the class of 2Bézout rings and the class of coherent rings are not comparable with the class of weakly Bézout rings.


Introduction
All rings considered below are commutative with unit and all modules are unital. A ring R is a Bézout ring if every finitely generated ideal of R is principal. Examples of Bézout rings are valuation rings, elementary divisor rings and Hermite rings. For instance see [6,10,11]. A ring R is called weakly Bézout if every finitely generated ideal of R contained in a principal proper ideal of R is itself principal (see [2,Definition 2 ]). If R is Bézout, then R is naturally weakly Bézout. Our aim in this paper is to prove that the converse is false in general.
For a nonnegative integer n, an R-module E is n-presented if there is an exact sequence of R-modules: where each F i is a finitely generated free R-module. In particular, 0-presented and 1-presented Rmodules are respectively, finitely generated and finitely presented R-modules.
A ring R is a coherent ring if every finitely generated ideal of R is finitely presented; equivalently, if (0 : a) and I ∩ J are finitely generated for every a ∈ R and every finitely generated ideals I and J of R [7, Theorem 2.3.2, p.45 ]. Examples of coherent rings are Noetherian rings, semihereditary rings and Bézout domains (see [7, p.47 ]).
Given nonnegative integers n and d, a ring R is called an (n, d)-ring if every n-presented R-module has projective dimension d; and a weak (n, d)-ring if every n-presented cyclic R-module has projective dimension d (equivalently, if every (n − 1)-presented ideal of R has projective dimension d − 1). See for instance [5,9].
A domain R is a Prüfer domain if every finitely generated ideal is projective (see [7, p.26 We say that R is a 2-Bézout ring if every finitely presented ideal of R is principal see [3]. This led us to consider the relation between the class of weakly Bézout rings and the class of 2-Bézout rings. Let A be a ring, E be an A-module and R := A ⋉ E be the set of pairs (a, e) with pairwise addition and multiplication given by: (a, e)(b, f ) = (ab, af + be). R is called the trivial ring extension of A by E. Considerable work, part of it summarized in Glaz's book [7] and Huckaba's book [8] where R is called the idealization of E by A, has been concerned with trivial ring extensions. These have proven to be useful in solving many open problems and conjectures for various contexts in (commutative and non-commutative) ring theory. See for instance [7,8,12].
In the context of rings containing regular elements, we show that the notion of weakly Bézout coincides with the definition of Bézout ring. The goal of this work is to exhibit a class of non-Bézout rings which are weakly Bézout rings. We show that the class of weakly Bézout rings is not stable under finite direct products. Also, we show that the class of 2-Bézout rings and the class of coherent rings are not comparable with the class of weakly Bézout rings. For this purpose, we study the transfer of this property to trivial ring extensions. We begin this section by giving a sufficient condition to have equivalence between Bézout and weakly Bézout properties.

1) If R is a Bézout ring, then R is a weakly Bézout ring. 2) Assume that R contains a non-invertible regular element (that is, R is not a total ring of quotients). Then, R is a Bézout ring if and only if R is a weakly Bézout ring.
Proof. 1) Clear.
2) It remains to show that, if R is a weakly Bézout ring and contains a regular element a non-invertible, then R is a Bézout ring. Let I be a finitely generated ideal of R and a a non-invertible regular element of R. Then aI ⊆ aR and so aI is principal since R is weakly Bézout. Thus I is principal since (I ∼ = aI), as desired.

Remark 2.3. By the above result, a non-Bézout ring which is a weakly Bézout ring is necessarily a total ring of quotient.
In this section, we study the possible transfer of the weakly Bézout property to various trivial extension contexts. First, we examine the context of trivial ring extensions of a local (A, M ) by an

1) R is a weakly Bézout ring if and only if so is
Proof. Assume that R is a weakly Bézout ring. Our aim is to show that A is weakly Bézout. Let I ⊆ J be two ideals of A such that I is finitely generated and J is principal proper. Then, I ⋉ 0 ⊆ J ⋉ 0 are two finitely generated proper ideals of R. Moreover, J ⋉ 0 = Aa ⋉ 0 = R(a, 0) for some element a of A and so I ⋉ 0 ⊆ J ⋉ 0 is a principal ideal of R since R is a weakly Bézout ring that is, Conversely, assume that A is a weakly Bézout ring. Our aim is to show that R is a weakly Bézout ring.
. . , n}, we which to show that I is principal. Three cases are then possible. Case 1 If b = 0. Then, a i = 0 for all i = 1, . . . , n; I := 0 ⋉ E 1 and J := 0 ⋉ E 2 where E 1 (resp.,E 2 ) is a vector subspace of E generated by the vectors e 1 , . . . , e n (resp., f ). Hence, E 1 is a (A/M )-vector space of rank at most 1 ( since . Also, we have the exact sequence of R-modules: Therefore, I ∼ = I 0 ⋉ 0 and since I 0 ⊆ J 0 ( because I ⊆ J), then I 0 = Aa for some element a ∈ A since A is a weakly Bézout ring and so I 0 ⋉ 0 = Aa ⋉ 0 = R(a, 0). Hence, I is a principal ideal of R in all cases. So, R is a weakly Bézout ring.
2) Assume that dim The condition dim (A/M) E = 1 is not sufficient in Theorem 2.4 (2) (see Example 2.8). Now, we are able to construct a non-Bézout ring which is a weakly Bézout ring.
Example 2.5. Let K be a field, E be a K-vector space such that dim K E 2 and R := K ⋉ E. Then:  Next, we explore a different context; namely, the trivial ring extension of a domain A by a K-vector space E, where K := qf (A). Proposition 2.9. Let A be a domain which is not a field, K := qf (A), E be a K-vector space and R := A ⋉ E be the trivial ring extension of A by E. If R is a weakly Bézout ring, then so is A and dim K E = 1.
Proof of Proposition 2.9. Assume that R is a weakly Bézout ring. We claim that A is weakly Bézout. Indeed, Let I := i=n i=1 Aa i for some positive integer n and J := Ab be two proper ideals of A such that I ⊆ J. Then, I ⋉ E := i=n i=1 Aa i ⋉ E = Aa 1 ⋉ E + Aa 2 ⋉ E + · · · + Aa n ⋉ E = R(a 1 , e 1 ) + R(a 2 , e 2 ) + · · · + R(a n , e n ) = i=n i=1 R(a i , 0) contained in J ⋉ E := Ab ⋉ E = R(b, 0). Therefore, I ⋉ E = R(a, k) = Aa ⋉ E for some element (a, k) of R since R is a weakly Bézout ring. Hence, I = Aa, and therefore A is a weakly Bézout ring. By way of contradiction, suppose that dim K E 2 and let {e, f } be a K-linearly independent set of E. Set I := R(0, e) + R(0, f ) ⊆ Aa ⋉ E. As in the proof of Theorem 2.4 (2), I is not a principal ideal of R, while I is contained in the principal ideal Aa ⋉ E.
Next, we give an example of non weakly Bézout ring.
Example 2.11. Let A be a Bézout domain which is not a field, K := qf (A). Then, the trivial ring extension of A by K 2 is not weakly Bézout by Proposition 2.9 .
It is straightforward to see that if a finite product R := n i=1 R i of commutative rings is a weakly Bézout ring, then R i is a weakly Bézout for every i, however a finite product of weakly Bézout rings is not necessarily a weakly Bézout ring as shown by the next example. Proof. It is clear that R 2 := A ⋉ E is a weakly Bézout ring by Theorem 2.4 (1). But, R 1 × R 2 is not a weakly Bézout ring. Indeed, let I be a principal proper ideal of R 1 . Then, I × (0 ⋉ E) is a non-principal finitely generated ideal of R 1 × R 2 contained in the principal ideal I × R 2 , which is a contradiction.
The following two examples show that is the class of weakly Bézout rings and the class of 2-Bézout rings are not comparable.  Proof. 1) R is Noetherian, then R is a coherent ring.
2) (X, Y ) is a finitely generated ideal of R which is not principal, then R is not a Bézout domain. Therefore, R is not a weakly Bézout domain, by Proposition 2.2.
Open Problem. Is the property weakly Bézout stable by homomorphic images and localizations ?