C-supplement submodules

In this study, cr-supplemented modules are introduced. Let Ñ module and S,Y≤Ñ be submodules. A module Ñ is cr-supplemented if, for each submodule S of Ñ, there is a submodule Y in Ñ such that Ñ=S+Y and S∩Y is crumbling. If each submodule of Ñ has c-supplement then Ñ is called cr-supplemented. Ñ is cr-supplemented if and only if ( Ñ)⁄(C(Ñ)) is semisimple. We provide characterization of rings for which their modules are cr-supplemented. With this, we show that cr-supplemented modules are closed under direct sums, submodules, homomorphic images and sums.