In this article, we study the class of rings in which every regular locally principal ideal is projective called LPP-rings. We investigate the transfer of this property to various constructions such as direct products, amalgamation of rings, and trivial ring extensions. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned property.
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 ⋈fJ, introduced by D'Anna, Finocchiaro and Fontana in . 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.