In this paper we introduce and investigate a class of those rings in which every principal ideal is finitely presented. We establish the transfer of this notion to the trivial ring extension, direct product and homomorphic image, and then generate new and original families of rings satisfying this property.

In this paper we introduce and investigate a class of those rings in which every projective ideal is regular. We establish the transfer of this notion to trivial ring extension, direct product, pullbacks, and amalgamation of rings along an ideal and then generate new and original families of rings satisfying this property.