Completely Slightly Compressible Modules

Cesim Celik

Abstract

In this note we introduce and investigate completely slightly compressible modules as a generalization of compressible modules. It is shown that if M is completely slightly compressible module and for any 0 \ne x ∈ M, xR is not isomorphic to any submodule of itself, then (1) every nonzero submodule of M contains a nonzero simple direct summand and (2) M has a decomposition M = M1⊕M2 where M1 is a semisimple submodule, M2 is a submodule which has an essential socle. Every simple submodule of a completely slightly compressible is a direct summand. Furthermore, an artinian completely slightly compressible ring is semisimple.