Nil$_{\ast}$-Noetherian rings

Abstract: In this paper, we say a ring $R$ is Nil$<em>{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$</em>{\ast}$-Noetherian rings, that is, $R$ is Nil$<em>{\ast}$-Noetherian if and only if $R[x]$ is Nil$</em>{\ast}$-Noetherian, if and only if $R[[x]]$ is Nil$<em>{\ast}$-Noetherian. Then we discuss some Nil$</em>{\ast}$-Noetherian properties on idealizations and bi-amalgamated algebras. Finally, we give the Cartan-Eilenberg-Bass Theorem for Nil$<em>{\ast}$-Noetherian rings in terms of Nil$</em>{\ast}$-injective modules and Nil$<em>{\ast}$-FP-injective modules. Besides, some examples are given to distinguish Nil$</em>{\ast}$-Noetherian rings, Nil$_{\ast}$-coherent rings and so on.