Rings in which every homomorphic image is a Noetherian domain

Abstract

In this paper, we introduce a generalization of the well-known notion of a Noetherian ring which we call a Q-Noetherian ring. We investigate the transfer of the Q-Noetherian property to trivial ring extensions, localizations, homomorphic image of rings, direct product of rings, and amalgamated algebras along an ideal. These results provide examples of non-Noetherian Q-Noetherian rings, of non-coherent Q-Noetherian rings and examples of non-Q-Noetherian coherent rings. The article includes a brief discussion of the scope and precision of our results.