Rings Whose Certain Modules are Dual Self-CS-Baer

Abstract

In this work, we characterize some rings in terms of dual self-CS-Baer modules (briefly, dsCS-Baer modules). We prove that any ring $\Sigma$ is a left and right artinian serial (briefly, as-) ring with $J^2=0$ iff $\Sigma\oplus Z$ is dsCS-Baer for every right $\Sigma$-module $Z$. If $\Sigma$ is a commutative ring, then we prove that $\Sigma$ is an as-ring iff $\Sigma$ is perfect and every $\Sigma$-module is a direct sum of (cyclic) dsCS-Baer $\Sigma$-modules. Also, we show that $\Sigma$ is a right perfect ring iff all countably generated free right $\Sigma$-modules are dsCS-Baer.