On $\jmath$-Artinian rings

Abstract

Let $R$ be a commutative ring with identity and $\jmath$ be an ideal of $R$. The aim of this paper is to introduce and study a new class of rings that is closely related to the class of Artinian rings. An ideal $I$ of $R$ is said to be a $\jmath$-ideal if $I \nsubseteq \jmath$. We say that $R$ is $\jmath$-Artinian if every descending chain of $\jmath$-ideals is stationary. We show that the theories of $\jmath$-Artinian rings resemble that of Artinian rings. We also provide several characterizations of $\jmath$-Artinian rings in terms of Artinian rings. Furthermore, we obtain some connections of $\jmath$-Artinian rings with semisimple rings, $\phi$-von Neumann regular rings, trivial ring extensions, and zero-dimensional rings.