Endo-principally projective modules

B. Ungor, N. Agayev, S. Halicioglu, A. Harmanci

Summary: Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S= End_R(M)$. 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 projective (or simply endo-p.p.) if for any $m\in M$, $l_S(m)=Se$ for some $e^2=e\in S$. 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.