Injective representations of quivers
Sang-Won Park,De-Ra Shin +1 more
TLDR
In this article, it was shown that a representation of a quiver is an injective representation if and only if it is isomorphic to a direct sum of representation of the types and where are injective left R-modules.Abstract:
We prove that is an injective representation of a quiver if and only if are injective left R-modules, is isomorphic to a direct sum of representation of the types and where are injective left R-modules. Then, we generalize the result so that a representation of a quiver is an injective representation if and only if each is an injective left R-module and the representation is a direct sum of injective representations.read more
Citations
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PROJECTIVE REPRESENTATIONS OF A QUIVER WITH THREE VERTICES AND TWO EDGES AS R(x)-MODULES
Juncheol Han,Sangwon Park +1 more
TL;DR: In this article, it was shown that the projective properties of representations of a quiver Q = • → • → → • as left R [ x ]-modules are not projective if P = 0.
Journal ArticleDOI
Rings described by some special morphisms of modules
TL;DR: The category of morphisms of R-modules as discussed by the authors is an extension of the category of R -modules using some special morphisms, such as phantom, Ext-phantom, flat and absolutely pure morphisms.
References
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A Homotopy of Quiver Morphisms with Applications to Representations
Edgar E. Enochs,Ivo Herzog +1 more
TL;DR: In this paper, it is shown that a morphism of quivers having a certain path lifting property has a decomposition that mimics the decomposition of maps of topological spaces into homotopy equivalences composed with fibrations.
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Projective representations of quivers
TL;DR: In this article, it was shown that P1 → f P2 is a projective representation of a quiver if and only if P1 and P2 are projective left R-modules, f is an injection, and f (P 1)⊂P 2 is a summand.
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A Homotopy of Quiver Morphisms with Applications to Representations
Edgar E. Enochs,Ivo Herzog +1 more