SEMIPRIME SUBMODULES OF GRADED MULTIPLICATION MODULES Lee, Sang-Cheol; Varmazyar, Rezvan;
Let G be a group. Let R be a G-graded commutative ring with identity and M be a G-graded multiplication module over R. A proper graded submodule Q of M is semiprime if whenever , where , n is a positive integer, and , then . We characterize semiprime submodules of M. For example, we show that a proper graded submodule Q of M is semiprime if and only if grad. Furthermore if M is finitely generated then we prove that every proper graded submodule of M is contained in a graded semiprime submodule of M. A proper graded submodule Q of M is said to be almost semiprime if (grad(Q)h(M))n(grad) = (Qh(M))n(grad). Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K M and for all , then we prove that Q + K is almost semiprime in M.