Rings for which every cosingular module is projective

Authors : Y. TALEBİ, A. R. M. HAMZEKOLAEE, M. HOSSEİNPOUR, A. HARMANCİ, B. UNGOR

Pages : 973-984

View : 17 | Download : 6

Publication Date : 2019-08-08

Article Type : Research Paper

Abstract :Let $R$ be a ring and $M$ be an $R$-module. In this paper we investigate modules $M$ such that every insert ignore into journalissuearticles values(simple); cosingular $R$-module is $M$-projective. We prove that every simple cosingular module is $M$-projective if and only if for $N\leq T\leq M$, whenever $T/N$ is simple cosingular, then $N$ is a direct summand of $T$. We show that every simple cosingular right $R$-module is projective if and only if $R$ is a right $GV$-ring. It is also shown that for a right perfect ring $R$, every cosingular right $R$-module is projective if and only if $R$ is a right $GV$-ring. In addition, we prove that if every $\delta$-cosingular right $R$-module is semisimple, then $\overline{Z}insert ignore into journalissuearticles values(M);$ is a direct summand of $M$ for every right $R$-module $M$ if and only if $\overline{Z}_{\delta}insert ignore into journalissuearticles values(M);$ is a direct summand of $M$ for every right $R$-module $M$.
Keywords : projective module, cosingular module, delta cosingular module, GV ring