On / - d -Primary Submodules

In this paper, we introduce and study a new class of submodules which unify the concepts of prime and primary submodules. Let M be a unital module over a commutative ring R ; / : L ð M Þ ! L ð M Þ [ ;fg be a reduction function and d : L ð R Þ ! L ð R Þ be an expansion function, where L ð M Þ is the lattice of all submodules of M and L ð R Þ is the lattice of all ideals of R : A proper submodule N of M is said to be a / - d -primary submodule of M if whenever am 2 N (cid:2) / ð N Þ for some a 2 R and m 2 M ; then either a 2 d ðð N : M ÞÞ or m 2 N : Many properties, characterizations and examples of / - d -primary submodules are given.