• Title, Summary, Keyword: primary submodule

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ON WEAKLY 2-ABSORBING PRIMARY SUBMODULES OF MODULES OVER COMMUTATIVE RINGS

  • Darani, Ahmad Yousefian;Soheilnia, Fatemeh;Tekir, Unsal;Ulucak, Gulsen
    • Journal of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1505-1519
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    • 2017
  • Assume that M is an R-module where R is a commutative ring. A proper submodule N of M is called a weakly 2-absorbing primary submodule of M if $0{\neq}abm{\in}N$ for any $a,b{\in}R$ and $m{\in}M$, then $ab{\in}(N:M)$ or $am{\in}M-rad(N)$ or $bm{\in}M-rad(N)$. In this paper, we extended the concept of weakly 2-absorbing primary ideals of commutative rings to weakly 2-absorbing primary submodules of modules. Among many results, we show that if N is a weakly 2-absorbing primary submodule of M and it satisfies certain condition $0{\neq}I_1I_2K{\subseteq}N$ for some ideals $I_1$, $I_2$ of R and submodule K of M, then $I_1I_2{\subseteq}(N:M)$ or $I_1K{\subseteq}M-rad(N)$ or $I_2K{\subseteq}M-rad(N)$.

ON PRIME SUBMODULES OF A FINITELY GENERATED PROJECTIVE MODULE OVER A COMMUTATIVE RING

  • Nekooei, Reza;Pourshafiey, Zahra
    • Communications of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.729-741
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    • 2019
  • In this paper we give a full characterization of prime submodules of a finitely generated projective module M over a commutative ring R with identity. Also we study the existence of primary decomposition of a submodule of a finitely generated projective module and characterize the minimal primary decomposition of this submodule. Finally, we characterize the radical of an arbitrary submodule of a finitely generated projective module M and study submodules of M which satisfy the radical formula.

ON DISTINGUISHED PRIME SUBMODULES

  • Cho, Yong-Hwan
    • Communications of the Korean Mathematical Society
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    • v.15 no.3
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    • pp.493-498
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    • 2000
  • In this paper we find some properties of distinguished prime submodules of modules and prove theorems about the dimension of modules.

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GRADED PRIMAL SUBMODULES OF GRADED MODULES

  • Darani, Ahmad Yousefian
    • Journal of the Korean Mathematical Society
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    • v.48 no.5
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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.

THE PRODUCT OF MULTIPLICATION SUBMODULES

  • ATANI, SHAHABADDIN EBRAHIMI
    • Honam Mathematical Journal
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    • v.27 no.1
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    • pp.1-8
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    • 2005
  • Let R be a commutative ring with non-zero identity. This paper is devoted to the study some of properties of the product of submodules of a multiplication module. Suppose N is a submodule of a multiplication R-module M. We give a condition which allows us to determine whether N is finitely generated when we assume some power of N is finitely generated.

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