Abstract
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 2- Bézout rings and class of coherent rings are not comparable with the class of weakly Bézout rings.
Highlights
We say that R is a 2- Bezout ring if every finitely presented ideal of R is principal see [3]
In the context of rings containing regular elements, we show that the notion of weakly Bezout coincides with the definition of Bezout ring
The goal of this work is to exhibit a class of non-Bezout rings which are weakly Bezout rings
Summary
Haitham El Alaoui and Hakima Mouanis abstract: In this paper, we examine the transfer of the proprety weakly Bezout to the trivial ring extensions. We say that R is a 2- Bezout ring if every finitely presented ideal of R is principal see [3]. A ring R is called a weakly Bezout ring if for every finitely generated ideals I and J of R satisfying that I ⊆ J R, when J is principal of R, so is I. Let I be a finitely generated ideal of R and a a non-invertible regular element of R. 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 Bezout ring that is, I ⋉ 0 = R(b, 0) = Ab ⋉ 0 for some element b of A. We assume that ((ai, ei)ni=1) is a minimal generating set of I, I0 :=
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