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