On simple submodules and C-rings

Abstract

In this paper; we will study the relationship between simple submodules of M and the C-ring. A ring R is called C-ring if every torsion module M over R has simple submodules. We investigated that if M is a torsion module over a semisimple ring; so, R is C-ring. Also, we study C-ring when R is Noetherian and T(M) = M such that any direct sum of modules with (SIP) has (SSP) implies that R is C- ring. We proved that if M is a torsion module and M = W + W′, this means R is C-ring where W and W′ are direct sum of M.