Nil$_{\ast}$-Artinian rings

Abstract

In this paper, we say a ring $R$ is Nil$_{\ast}$-Artinian if any descending chain of nil ideals stabilizes. We first study Nil$_{\ast}$-Artinian properties in terms of quotients, localizations, polynomial extensions and idealizations, and then study the transfer of Nil$_{\ast}$-Artinian rings to amalgamated algebras. Besides, some examples are given to distinguish Nil$_{\ast}$-Artinian rings, Nil$_{\ast}$-Noetherian rings and Nil$_{\ast}$-coherent rings.