Modules whose closed \(M\)-cyclics are summand. (English) Zbl 1219.16005
Summary: We introduce the concept of CMS modules. A right \(R\)-module \(M\) is called CMS if every closed \(M\)-cyclic submodule of \(M\) is a direct summand. An example of CMS module which is not CS is given. We characterize semi-simple Artinian rings in terms of CMS modules. We prove that over a right hereditary ring \(R\), every projective right \(R\)-module is a CMS module.
MSC:
16D50 | Injective modules, self-injective associative rings |
16D40 | Free, projective, and flat modules and ideals in associative algebras |
16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |