• 대한수학회지
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• 제49권2호
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• pp.435-447
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• 2012

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

• 대한수학회지
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• 제48권5호
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• pp.927-938
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• 2011

Let G be an abelian monoid with identity e. Let R be a G-graded commutative ring, and M a graded R-module. In this paper we first introduce the concept of graded primal submodules of M an give some basic results concerning this class of submodules. Then we characterize the graded primal ideals of the idealization R(+)M.

• 대한수학회논문집
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• 제38권4호
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• pp.993-999
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• 2023

Let A be a G-graded commutative ring with identity and M a graded A-module. Let m, n be positive integers with m > n. A proper graded submodule L of M is said to be graded (m, n)-closed if a^m_g·x_t ∈ L implies that aⁿ_g·x_t ∈ L, where a_g ∈ h(A) and x_t ∈ h(M). The aim of this paper is to explore some basic properties of these class of submodules which are a generalization of graded (m, n)-closed ideals. Also, we investigate GC^m_n - rad property for graded submodules.

• Korean Journal of Mathematics
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• 제20권3호
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• pp.315-321
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• 2012

In this paper, we provide the necessary and sufficient conditions for a faithful graded module to be a graded multiplication module and for a graded submodule of a faithful gr-multiplication to be gr-essential.