Semiprime submodules of graded multiplication modules
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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 InK ⊆ Q,where I ⊆ h(R), n is a positive integer, and K h(M), then IK⊆Q. 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(Q) ∩ h(M) = Q ∩ h(M). 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(0M) ∩ h(M))= (Q ∩ h(M))n(grad(0M) ∩ Q ∩ h(M)):Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K ̸= M and Q ∩ K Mg for all g 2 G, then we prove that Q + K is almost semiprime in M.
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 InK ⊆ Q,where I ⊆ h(R), n is a positive integer, and K h(M), then IK⊆Q. 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(Q) ∩ h(M) = Q ∩ h(M). 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(0M) ∩ h(M))= (Q ∩ h(M))n(grad(0M) ∩ Q ∩ h(M)):Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K ̸= M and Q ∩ K Mg for all g 2 G, then we prove that Q + K is almost semiprime in M.
참고문헌 (6)
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