• 제목/요약/키워드: projective module

검색결과 88건 처리시간 0.027초

OPENLY SEMIPRIMITIVE PROJECTIVE MODULE

  • Bae, Soon-Sook
    • 대한수학회논문집
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    • 제19권4호
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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.

PROJECTIVE PROPERTIES OF REPRESENTATIONS OF A QUIVER Q = • → • AS R[x]-MODULES

  • Park, Sangwon;Kang, Junghee;Han, Juncheol
    • Korean Journal of Mathematics
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    • 제18권3호
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    • pp.243-252
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    • 2010
  • In this paper we extend the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left R-modules to the projective properties of representations of quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\rightarrow}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. And we show $0{\rightarrow}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module if and only if $0{\rightarrow}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. Then we show if P is a projective left R-module then $R[x]\longrightarrow^{id}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. We also show that if $L\longrightarrow^{id}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module, then $L[x]\longrightarrow^{id}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules.

A Characterization of Nonnil-Projective Modules

  • Hwankoo Kim;Najib Mahdou;El Houssaine Oubouhou
    • Kyungpook Mathematical Journal
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    • 제64권1호
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    • pp.1-14
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    • 2024
  • Recently, Zhao, Wang, and Pu introduced and studied new concepts of nonnil-commutative diagrams and nonnil-projective modules. They proved that an R-module that is nonnil-isomorphic to a projective module is nonnil-projective, and they proposed the following problem: Is every nonnil-projective module nonnil-isomorphic to some projective module? In this paper, we delve into some new properties of nonnil-commutative diagrams and answer this problem in the affirmative.

(CO)RETRACTABILITY AND (CO)SEMI-POTENCY

  • Hakmi, Hamza
    • Korean Journal of Mathematics
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    • 제25권4호
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    • pp.587-606
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    • 2017
  • This paper is a continuation of study semi-potentness endomorphism rings of module. We give some other characterizations of endomorphism ring to be semi-potent. New results are obtained including necessary and sufficient conditions for the endomorphism ring of semi(injective) projective module to be semi-potent. Finally, we characterize a module M whose endomorphism ring it is semi-potent via direct(injective) projective modules. Several properties of the endomorphism ring of a semi(injective) projective module are obtained. Besides to that, many necessary and sufficient conditions are obtained for semi-projective, semi-injective modules to be semi-potent and co-semi-potent modules.

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

  • Nekooei, Reza;Pourshafiey, Zahra
    • 대한수학회논문집
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    • 제34권3호
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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.

PROJECTIVE REPRESENTATIONS OF A QUIVER WITH THREE VERTICES AND TWO EDGES AS R[x]-MODULES

  • Han, Juncheol;Park, Sangwon
    • Korean Journal of Mathematics
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    • 제20권3호
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    • pp.343-352
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    • 2012
  • In this paper we show that the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\longrightarrow}0{\longrightarrow}P[x]$ is a projective representation of a quiver Q as $R[x]$-modules, but $P[x]{\longrightarrow}0{\longrightarrow}0$ is not a projective representation of a quiver Q as $R[x]$-modules, if $P{\neq}0$. And we show a representation $0{\longrightarrow}P[x]\longrightarrow^{id}P[x]$ of a quiver Q is a projective representation, if P is a projective left R-module, but $P[x]\longrightarrow^{id}P[x]{\longrightarrow}0$ is not a projective representation of a quiver Q as $R[x]$-modules, if $P{\neq}0$. Then we show a representation $P[x]\longrightarrow^{id}P[x]\longrightarrow^{id}P[x]$ of a quiver Q is a projective representation, if P is a projective left R-module.

Almost Projective Modules over Artin Algebras

  • Park, Jun Seok
    • 충청수학회지
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    • 제1권1호
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    • pp.43-53
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    • 1988
  • The main result of this paper is a characterization of almost projective modules over art in algebras by means of irreducible maps and almost split sequences. A module X is an almost projective module if and only if it has a presentation $0{\longrightarrow}L{\longrightarrow^{\alpha}}P{\longrightarrow}X{\longrightarrow}0$ with projective module P and irreducible maps ${\alpha}$. Let X be an injective almost projective non simple module and $0{\rightarrow}Dtr(x){\rightarrow}E{\rightarrow}X{\rightarrow}0$ be an almost split sequence. If $E=E_1{\oplus}E_2$ is a direct decomposition of indecomposable modules then ${\ell}(X)=3$.

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GORENSTEIN PROJECTIVE DIMENSIONS OF COMPLEXES UNDER BASE CHANGE WITH RESPECT TO A SEMIDUALIZING MODULE

  • Zhang, Chunxia
    • 대한수학회보
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    • 제58권2호
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    • pp.497-505
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    • 2021
  • Let R → S be a ring homomorphism. The relations of Gorenstein projective dimension with respect to a semidualizing module of homologically bounded complexes between U ⊗LR X and X are considered, where X is an R-complex and U is an S-complex. Some sufficient conditions are given under which the equality ${\mathcal{GP}}_{\tilde{C}}-pd_S(S{\otimes}{L \atop R}X)={\mathcal{GP}}_C-pd_R(X)$ holds. As an application it is shown that the Auslander-Buchsbaum formula holds for GC-projective dimension.

ON 𝜙-EXACT SEQUENCES AND 𝜙-PROJECTIVE MODULES

  • Zhao, Wei
    • 대한수학회지
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    • 제58권6호
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    • pp.1513-1528
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    • 2021
  • Let R be a commutative ring with prime nilradical Nil(R) and M an R-module. Define the map 𝜙 : R → RNil(R) by ${\phi}(r)=\frac{r}{1}$ for r ∈ R and 𝜓 : M → MNil(R) by ${\psi}(x)=\frac{x}{1}$ for x ∈ M. Then 𝜓(M) is a 𝜙(R)-module. An R-module P is said to be 𝜙-projective if 𝜓(P) is projective as a 𝜙(R)-module. In this paper, 𝜙-exact sequences and 𝜙-projective R-modules are introduced and the rings over which all R-modules are 𝜙-projective are investigated.