SEMIPRIME SUBMODULES OF GRADED MULTIPLICATION MODULES

Abstract

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 <TEX>$I^nK{\subseteq}Q$</TEX>, where <TEX>$I{\subseteq}h(R)$</TEX>, n is a positive integer, and <TEX>$K{\subseteq}h(M)$</TEX>, then <TEX>$IK{\subseteq}Q$</TEX>. 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<TEX>$(Q){\cap}h(M)=Q+{\cap}h(M)$</TEX>. 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)<TEX>$\cap$</TEX>h(M))n(grad<TEX>$(0_M){\cap}h(M)$</TEX>) = (Q<TEX>$\cap$</TEX>h(M))n(grad<TEX>$(0_M){\cap}Q{\cap}h(M)$</TEX>). Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K <TEX>$\neq$</TEX> M and <TEX>$Q{\cap}K{\subseteq}M_g$</TEX> for all <TEX>$g{\in}G$</TEX>, then we prove that Q + K is almost semiprime in M.