On the Restricted Minimum Condition for Rings

Abstract

Generalizing Artinian rings, a ring R is said to have right restricted minimum condition (${\mathrm{r.RMC}}$, for short) if R/A is an Artinian right R-module for any essential right ideal A of R. It is asked in Jain et al. [Cyclic Modules and the Structure of Rings, Oxford University Press, Oxford, 2012, 3.17 Questions (2)] that (i) Is a left self-injective ring with ${\mathrm{r.RMC}}$ quasi-Frobenius? (ii) Whether a serial ring with ${\mathrm{r.RMC}}$ must be Noetherian? We carry out a study of rings with ${\mathrm{r.RMC}}$ and determine when a right extending ring has ${\mathrm{r.RMC}}$ in terms of rings ${\begin{bmatrix} S&{}\quad M\\ 0&{}\quad R\end{bmatrix}}$ such that S is right Artinian, $M_{Q}$ is semisimple ($Q={\mathrm{Q}}(R)$) and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with ${\mathrm{r.RMC}}$ is quasi-Frobenius if and only if $\hbox {Z}_{r}(R) = \hbox {Z}_{l}(R)$ if and only if $\hbox {Z}_{r}(R)$ is a finitely generated left ideal and ${\mathrm{N}}(R)\cap {\mathrm{Soc}}(R_{R})$ is a finitely generated right ideal. Right serial rings with ${\mathrm{r.RMC}}$ are studied and proved that a non-singular serial ring has ${\mathrm{r.RMC}}$ if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that $\mathrm{RMC}$ is not symmetric for a ring.