Endo-principally Projective Modules

Abstract

Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). In this paper, we introduce a class of modules that is a generalization of principally projective (or simply p.p.) rings and Baer modules. The module M is called endo-principally pro- jective (or simply endo-p.p.) if for any m 2 M, lS(m) = Se for some e2 = e 2 S. For an endo-p.p. module M, we prove that M is endo- rigid (resp., endo-reduced, endo-symmetric, endo-semicommutative) if and only if the endomorphism ring S is rigid (resp., reduced, symmetric, semicommutative), and we also prove that the module M is endo-rigid if and only if M is endo-reduced if and only if M is endo-symmetric if and only if M is endo-semicommutative if and only if M is abelian. Among others we show that if M is abelian, then every direct summand of an endo-p.p. module is also endo-p.p. AMS Mathematics Subject Classi cation (2010): 13C99, 16D80, 16U80. Key words and phrases: Baer modules, quasi-Baer modules, endo-princi- pally quasi-Baer modules, endo-p.p. modules, endo-symmetric modules, endo-reduced modules, endo-rigid modules, endo-semicommutative mod- ules, abelian modules.