Z-primary submodules

Abstract EN

Let R is a commutative ring with identity and M be a unitary R -module , a submodule N of an R-module M is called primary if whenever rx E N ; r E R,x E M, then either x E Nor r" 6 [N\ M] for some n 6 Z+.

In this paper we say that N is Z -primary submodule of an R -module M if whenever f(x).x E N,f E M* = Hom(M,R), xe M, then either x E N or (f(x))n 6 [N:Mf for some n E Z+, where [N: M] = {r: r E R and rM £ N}.

We also study a Z-primary module if (0) is a Z-primary submodule of M.

We give many properties of Z-primary submodule of M and Z-primary module.