Max(Min)C11 Modules with Their Endomorphism Rings

Sarapee Chairat

Abstract

In this paper, we consider the class of rings and modules with extending properties, and study an intensive class of max(min)C11 modules together with their endomorphism rings. An R−moduleM is maxC11 module if every maximal submodule with nonzero left annihilator has a complement which is a direct summand of M. M is called a minC11 if every minimal submodule has a complement which is a direct summand of M. We prove that if M is a finitely generated, quasi-projective self-generator, thenM is C11 (resp. maxC11 , minC11 , max-minC11) module if and only if its endomorphism ring S is a rightC11 (resp. maxC11, minC11, max-minC11) ring. If M is a prime module, then M is nonsingular, max-minC11 with a uniform submodule if and only if S is right and left nonsingular, right and left max-minC11with uniform right and left ideals. Moreover, if M is a semiprime, weak duo module, then M is maxC11if and only if it is minC11.