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.