AI Summary & Key Takeaways
AI InsightIn addition, we prove that if every $\delta$-cosingular right $R$-module is semisimple, then $\overline{Z}(M)$ is a direct summand of $M$ for every right $R$-module $M$ if and only if $\overline{Z}_{\delta}(M)$ is a direct summand of $M$ for every right $R$-module $M$.
📌 3 Key Takeaways
Abstract
Let $R$ be a ring and $M$ be an $R$-module. In this paper we investigate modules $M$ such that every (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 $\overline{Z}(M)$ is a direct summand of $M$ for every right $R$-module $M$ if and only if $\overline{Z}_{\delta}(M)$ is a direct summand of $M$ for every right $R$-module $M$.