ON GRADED Ie-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

Let G be a group with identity e. Let R be a G-graded commutative ring with identity and M a graded R-module. We introduce the concept of graded Ie-prime submodule as a generalization of a graded prime submodule for I = circle plus(g is an element of G) I-g a fixed graded ideal of R. We give a number of results concerning this class of graded submodules and their homogeneous components. A proper graded submodule N of M is said to be a graded I-e-prime submodule of M if whenever r(g) is an element of h(R) and m(h) is an element of h(M) with r(g)m(h) is an element of N - IeN, then either r(g) 2 (N : R M) or m(h) is an element of N.