Nil$_{\ast}$-Noetherian rings

TL;DR

This paper introduces Nil$_{ ext{*}}$-Noetherian rings, explores their properties, and extends classical theorems like Hilbert basis to this new class, including applications to algebraic constructions and module theory.

Contribution

It defines Nil$_{ ext{*}}$-Noetherian rings, proves the Hilbert basis theorem for them, and establishes a Cartan-Eilenberg-Bass theorem in this context, expanding the theory of Noetherian-like rings.

Findings

Hilbert basis theorem holds for Nil$_{ ext{*}}$-Noetherian rings

Characterization of Nil$_{ ext{*}}$-Noetherian property in polynomial and power series rings

Development of the Cartan-Eilenberg-Bass theorem for these rings

Abstract

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.