• Title/Summary/Keyword: dense submodule

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A NOTE ON MONOFORM MODULES

  • Hajikarimi, Alireza;Naghipour, Ali Reza
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.2
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    • pp.505-514
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    • 2019
  • Let R be a commutative ring with identity and M be a unitary R-module. A submodule N of M is called a dense submodule if $Hom_R(M/N,\;E_R(M))=0$, where $E_R(M)$ is the injective hull of M. The R-module M is said to be monoform if any nonzero submodule of M is a dense submodule. In this paper, among the other results, it is shown that any kind of the following module is monoform. (1) The prime R-module M such that for any nonzero submodule N of M, $Ann_R(M/N){\neq}Ann_R(M)$. (2) Strongly prime R-module. (3) Faithful multiplication module over an integral domain.

ON MULTIPLICATION MODULES (II)

  • Cho, Yong-Hwan
    • Communications of the Korean Mathematical Society
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    • v.13 no.4
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    • pp.727-733
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    • 1998
  • In this short paper we shall find some properties on multiplication modules and prove three theorems.

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SOME ASPECTS OF ZARISKI TOPOLOGY FOR MULTIPLICATION MODULES AND THEIR ATTACHED FRAMES AND QUANTALES

  • Castro, Jaime;Rios, Jose;Tapia, Gustavo
    • Journal of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1285-1307
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    • 2019
  • For a multiplication R-module M we consider the Zariski topology in the set Spec (M) of prime submodules of M. We investigate the relationship between the algebraic properties of the submodules of M and the topological properties of some subspaces of Spec (M). We also consider some topological aspects of certain frames. We prove that if R is a commutative ring and M is a multiplication R-module, then the lattice Semp (M/N) of semiprime submodules of M/N is a spatial frame for every submodule N of M. When M is a quasi projective module, we obtain that the interval ${\uparrow}(N)^{Semp}(M)=\{P{\in}Semp(M){\mid}N{\subseteq}P\}$ and the lattice Semp (M/N) are isomorphic as frames. Finally, we obtain results about quantales and the classical Krull dimension of M.