Nil$_{\ast}$-Noetherian rings

Abstract

In this paper, we say a ring R is Nil * -Noetherian provided that any nil ideal is finitely generated. First, we show that Hilbert basis theorem holds for Nil * -Noetherian rings, and then we characterize the Nil * -Noetherian property of the idealization. After computing the nil-radical of the bi-amalgamated algebras under some assumptions, we finally characterize Nil * -Noetherian rings under some bi-amalgamated constructions. Besides, some examples are given to distinguish Nil * -Noetherian rings, Nil * -coherent rings and so on.