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In this paper, we introduce a generalization of the well -known notion of a P-Bezout rings and a 2-Bezout rings, which we call a P-2-Bezout rings. We establish the transfer of this notion to trivial ring extensions and to pullbacks. There results provide example of non 2-Bezout ring which is P-2-Bezout ring and example of non P-Bezout ring which is P-2-Bezout ring. Our aim is to provide new classe of commutative rings.

In this paper, we introduce a generalization of the well-known notion of weak coherent rings, which we call weak [Formula: see text]-coherent rings. We investigate the stability of this property under direct products and homomorphic image, and its transfer to various contexts of constructions such as trivial ring extensions and amalgamated algebras along an ideal. We conclude with various examples of weak [Formula: see text]-coherent 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 2- Bézout rings and class of coherent rings are not comparable with the class of weakly Bézout rings.

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