Endomorphism rings of modules over prime rings

TL;DR

This paper investigates conditions under which endomorphism rings of *-prime modules over prime rings are prime, exploring theoretical criteria and potential counterexamples in ring and module theory.

Contribution

It introduces conditions ensuring the primeness of endomorphism rings of *-prime modules and discusses the possibility of non-prime endomorphism rings.

Findings

Several conditions guarantee the primeness of endomorphism rings

Contours of a potential counterexample are outlined

Theoretical criteria for primeness are established

Abstract

Endomorphism rings of modules appear as the center of a ring, as the fix ring of ring with group action or as the subring of constants of a derivation. This note discusses the question whether certain *-prime modules (introduced by Bican et al.) have a prime endomorphism ring. Several conditions are presented that guarantee the primness of the endomorphism ring. The contours of a possible example of a *-prime module whose endomorphism ring is not prime are traced.