• Title/Summary/Keyword: projective

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PURE-DIRECT-PROJECTIVE OBJECTS IN GROTHENDIECK CATEGORIES

  • Batuhan Aydogdu;Sultan Eylem Toksoy
    • Honam Mathematical Journal
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    • v.45 no.2
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    • pp.269-284
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    • 2023
  • In this paper we study generalizations of the concept of pure-direct-projectivity from module categories to Grothendieck categories. We examine for which categories or under what conditions pure-direct-projective objects are direct-projective, quasi-projective, pure-projective, projective and flat. We investigate classes all of whose objects are pure-direct-projective. We give applications of some of the results to comodule categories.

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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    • v.18 no.3
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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.

SOME RESULTS ON 2-STRONGLY GORENSTEIN PROJECTIVE MODULES AND RELATED RINGS

  • Dong Chen;Kui Hu
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.4
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    • pp.895-903
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    • 2023
  • In this paper, we give some results on 2-strongly Gorenstein projective modules and related rings. We first investigate the relationship between strongly Gorenstein projective modules and periodic modules and then give the structure of modules over strongly Gorenstein semisimple rings. Furthermore, we prove that a ring R is 2-strongly Gorenstein hereditary if and only if every ideal of R is Gorenstein projective and the class of 2-strongly Gorenstein projective modules is closed under extensions. Finally, we study the relationship between 2-Gorenstein projective hereditary and 2-Gorenstein projective semisimple rings, and we also give an example to show the quotient ring of a 2-Gorenstein projective hereditary ring is not necessarily 2-Gorenstein projective semisimple.

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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    • v.20 no.3
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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.

CONVEX DECOMPOSITIONS OF REAL PROJECTIVE SURFACES. III : FOR CLOSED OR NONORIENTABLE SURFACES

  • Park, Suh-Young
    • Journal of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.1139-1171
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    • 1996
  • The purpose of our research is to understand geometric and topological aspects of real projective structures on surfaces. A real projective surface is a differentiable surface with an atlas of charts to $RP^2$ such that transition functions are restrictions of projective automorphisms of $RP^2$. Since such an atlas lifts projective geometry on $RP^2$ to the surface locally and consistently, one can study the global projective geometry of surfaces.

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DING PROJECTIVE MODULES WITH RESPECT TO A SEMIDUALIZING MODULE

  • Zhang, Chunxia;Wang, Limin;Liu, Zhongkui
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.339-356
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    • 2014
  • In this paper, we introduce and discuss the notion of $D_C$-projective modules over commutative rings, where C is a semidualizing module. This extends Gillespie and Ding, Mao's notion of Ding projective modules. The properties of $D_C$-projective dimensions are also given.

OPENLY SEMIPRIMITIVE PROJECTIVE MODULE

  • Bae, Soon-Sook
    • 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.

REALIZING A FAKE PROJECTIVE PLANE AS A DEGREE 25 SURFACE IN ℙ5

  • Lev Borisov;Zachary Lihn
    • Journal of the Korean Mathematical Society
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    • v.61 no.4
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    • pp.683-692
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    • 2024
  • Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in ℙ9. In this paper, we study Keum's fake projective plane (a = 7, p = 2, {7}, D327) and use the equations of [1] to construct an embedding of fake projective plane in ℙ5. We also simplify the 84 cubic equations defining the fake projective plane in ℙ9.

COMPLETELY RIGHT PROJECTIVE SEMIGROUPS

  • Moon, Eunho-L.
    • The Pure and Applied Mathematics
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    • v.9 no.2
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    • pp.119-128
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    • 2002
  • We here characterize semigroups (which are called completely right projective semigroups) for which every S-automaton is projective, and then examine some of the relationships with the semigroups (which are called completely right injective semigroups) in which every S-automaton is injective.

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