On (A)-rings and strong (A)-rings issued from amalgamations

Abstract

A ring R has the ( A )-property (resp., strong ( A )-property) if every finitely generated ideal of R consisting entirely of zero divisors (resp., every finitely generated ideal of R generated by a finite number of zero-divisors elements of R) has a nonzero annihilator. The class of commutative rings with property ( A ) is quite large; for example, Noetherian rings, rings whose prime ideals are maximal, the polynomial ring R [ x ] and rings whose total ring of quotients are von Neumann regular. Let f : A → B be a ring homomorphism and J be an ideal of B . In this paper, we investigate when the ( A )-property and strong ( A )-property are satisfied by the amalgamation of rings denoted by A ⋈ f J , introduced by D'Anna, Finocchiaro and Fontana in [3]. Our aim is to construct new original classes of ( A )-rings that are not strong ( A )-rings, ( A )-rings that are not Noetherian and ( A )-rings whose total ring of quotients are not Von Neumann regular rings.