Abstract
For a group G with identity e and a commutative G-graded ring R with a nonzero unity 1, we introduce the concepts of graded r-submodules and graded special r-submodules, which are generalizations for the notion of graded r-ideals. For a nonzero G-graded R-module IV, a proper graded R-submodule K of M is said to be a graded r-submodule (resp., a graded special r-submodule) if whenever a is an element of h(R) and x is an element of h(M) such that ax is an element of K with Ann(M) (a) = {0} (resp., Ann(R) (x)= {0}), then x is an element of K (resp., a is an element of (K :(R) M)). We study various properties of graded r-submodules and graded special r-submodules, and we give several examples of these two new classes of graded modules.