Rings which are Essential over their Centers, II

TL;DR

This paper investigates the properties of centrally essential rings, especially exterior algebras over modules, and explores semi-Artinian centrally essential rings, expanding understanding of their structure beyond commutative cases.

Contribution

It characterizes centrally essential exterior algebras over finitely generated free modules and studies properties of semi-Artinian centrally essential rings.

Findings

Exterior algebras are centrally essential under certain conditions.

Semi-Artinian centrally essential rings exhibit specific structural properties.

Extension of modules plays a key role in defining central essentiality.

Abstract

A ring $R$ with center $C$ is said to be\textit{centrally essential} if the module $R_C$ is an essential extension of the module $C_C$. We describe centrally essential exterior algebras of finitely generated free modules over not necessary commutative rings and study properties of semi-Artinian centrally essential rings. V.T.Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A.~ A.A. Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.