• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.289-297
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• 2010

By using as main tool the theory of prime submodules this article is devoted to describing the structure of the minimal prime ideals of the equidimensional symmetric algebra of a finitely generated module.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.475-484
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• 2023

Let R be a commutative ring with identity. In this paper, we characterize the prime submodules of a free R-module F of finite rank with at most n generators, when R is a GCD domain. Also, we show that if R is a Bézout domain, then every prime submodule with n generators is the row space of a prime matrix. Finally, we study the existence of primary decomposition of a submodule of F over a Bézout domain and characterize the minimal primary decomposition of this submodule.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1129-1147
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• 2021

This paper aims to analyze the PTG module for the finite-dimensional odd Contact superalgebra over a field of prime characteristic by using the method of Hu and Shen's mixed product realization. The general acting law in odd Contact superalgebra is obtained. In addition, the structure and irreducibility of graded module for odd Contact superalgebra are discussed.

• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.233-242
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• 2004

The aim of this note is to study properties of the generalized centroid of the semi-prime gamma rings. Main results are the following theorems: (1) Let M be a semi-prime $\Gamma$-ring and Q a quotient $\Gamma$-ring of M. If W is a non-zero submodule of the right (left) M-module Q, then $W\Gamma$W $\neq 0. Furthermore Q is a semi-prime $\Gamma$-ring. (2) Let M be a semi-prime $\Gamma$-ring and $C_{{Gamma}$ the generalized centroid of M. Then $C_{\Gamma}$ is a regular $\Gamma$-ring. (3) Let M be a semi-prime $\Gamma$-ring and $C_{\gamma}$ the extended centroid of M. If $C_{\gamma}$ is a $\Gamma$-field, then the $\Gamma$-ring M is a prime $\Gamma$-ring.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.553-563
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• 2017

Different topological dimensions related to the pseudo-prime spectrum of topological modules are studied. An example of topological modules is introduced. Also, we give a result about Noetherianness of the pseudo-prime spectrum of topological modules.

• Communications of the Korean Mathematical Society
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• v.19 no.4
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• pp.619-637
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• 2004

In this paper, a left module over an associative ring with identity is defined to be openly semiprimitive (strongly semiprimitive, respectively) by the zero intersection of all maximal open fully invariant submodules (all maximal open submodules which are fully invariant, respectively) of it. For any projective module, the openly semiprimitivity of the projective module is an equivalent condition of the semiprimitivity of endomorphism ring of the projective module and the strongly semiprimitivity of the projective module is an equivalent condition of the endomorphism ring of the projective module being a sub direct product of a set of subdivisions of division rings.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.1053-1066
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• 2010

In this paper, several properties of endomorphism rings of modules are investigated. A multiplication module M over a commutative ring R induces a commutative ring $M^*$ of endomorphisms of M and hence the relation between the prime (maximal) submodules of M and the prime (maximal) ideals of $M^*$ can be found. In particular, two classes of ideals of $M^*$ are discussed in this paper: one is of the form $G_{M^*}\;(M,\;N)\;=\;\{f\;{\in}\;M^*\;|\;f(M)\;{\subseteq}\;N\}$ and the other is of the form $G_{M^*}\;(N,\;0)\;=\;\{f\;{\in}\;M^*\;|\;f(N)\;=\;0\}$ for a submodule N of M.

• Journal of the Chungcheong Mathematical Society
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• v.18 no.1
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• pp.87-93
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• 2005

For a left R-module M, we identify certain submodules of M that play a role analogous to that of ideals in the ring R. We investigate some properties of M-ideals in the submodules of M and also study Jacobson radicals of a submodule of M.

• Kyungpook Mathematical Journal
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• v.56 no.4
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• pp.1085-1101
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• 2016

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each $m{\in}M$ and elements a, $b{\in}R$, $abm{\in}N$ implies that $am{\in}N$ or $bm{\in}N$. We introduce the concept of "weakly classical prime submodules" and we will show that this class of submodules enjoys many properties of weakly 2-absorbing ideals of commutative rings. A proper submodule N of M is a weakly classical prime submodule if whenever $a,b{\in}R$ and $m{\in}M$ with $0{\neq}abm{\in}N$, then $am{\in}N$ or $bm{\in}N$.

• Honam Mathematical Journal
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• v.2 no.1
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• pp.13-18
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• 1980

모든 기호(記號)는 G.E Bredon의 저(著) Sheaf Theory의 기호(記號)를 따른다. A가 torsion free이며 elementary sheaf이라 하자. 그리고 L을 injective L-module이라 하자 $dim_{\varphi}X<{\infty}$이라면 support의 $family{\varphi}$와 locally subset z에 대하여 ${\Gamma}_{z}(^{\sim}Hom({\Gamma}_{\varphi}(L),L){\otimes}A){\simeq}H_0{^{z}}(X:A)\;H_{-p}{^{z}}(X:A)=0,\;p=1,2,3,$⋯⋯ 이며 support의 family c와 compact subset z에 대하여도 ${\Gamma}_{z}(^{\sim}Hom({\Gamma}_{c}(L),L){\otimes}A){\simeq}H_0{^{z}}(X:A)\;H_{-y}{^{z}}(X:A)=0,\;p=1,2,3,$⋯⋯ A가 elementary이면 locally closed z와 z에서 closed인 $z^{\prime}$ 그리고 $z^{\prime\prime}=z-z^{\prime}$에 대하여 exact sequence ⋯⋯${\rightarrow}H^{z^{\prime}}_{p}\;(X:A){\rightarrow}H^{z}_{p}(X:A){\rightarrow}H^{z^{\prime\prime}}_{p}\;(X:A){\rightarrow}$⋯⋯ 가 존재(存在)한다.