Ideals and Factor Rings of Centrally Essential Rings

TL;DR

This paper investigates centrally essential rings, showing that such rings are not always right quasi-invariant, and provides conditions under which all maximal right ideals are actual ideals.

Contribution

It introduces the concept of central essentiality in rings and establishes conditions ensuring maximal right ideals are ideals, expanding understanding of ring structure.

Findings

Centrally essential rings are not necessarily right quasi-invariant.

Conditions are identified that guarantee all maximal right ideals are ideals.

Central essentiality can be used to analyze ideal properties in rings.

Abstract

It is proved that the ring $R$ with center $Z(R)$, such that the module $R_{Z(R)}$ is an essential extension of the module $Z(R)_{Z(R)}$, is not necessarily right quasi-invariant, i.e., maximal right ideals of the ring $R$ are not necessarily ideals. We use central essentiality to obtain conditions which are sufficient to the property that all maximal right ideals are ideals