A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left Artinian and QF-$3$

Abstract:

A module $M$ is called a CS module if every submodule of $M$ is essential in a direct summand of $M$. A ring $R$ is said to be right (countably) $\Sigma$-CS if any direct sum of (countably many) copies of the right $R$-module $R$ is CS. It is shown that for a right countably $\Sigma$-CS ring $R$ the following are equivalent: (i) $R$ is right $\Sigma$-CS, (ii) $R$ has ACC or DCC on projective principal right ideals, (iii) $R$ has finite right uniform dimension and ACC or DCC holds on projective uniform principal right ideals of $R$, (iv) $R$ is semiperfect. From results of Oshiro [12], [13], under these conditions, $R$ is left artinian and QF-$3$. As a consequence, a ring $R$ is quasi-Frobenius if it is right countably $\Sigma$-CS, semiperfect and no nonzero projective right ideals are contained in its Jacobson radical.