A new approach to multiplication modules via (delta)-small submodules

"Let R be a commutative ring and M an R-module. In this work we introduce two new generalizations of multiplication modules via delta-small submodules and small submodules of a fixed module. A module M is said to be (delta)-small multiplication provided for every (delta-)small submodule of N of M, there is an ideal I of R such that N=IM. We study some general properties of both delta-small multiplication modules and also small multiplication modules. A counterexample is presented to state this fact that the class of all delta-small multiplication modules lies exactly between the class of multiplication modules and small multiplication modules. We show that any direct summand of a (delta)-small multiplication module inherits the property."