Filomat 2023 Volume 37, Issue 25, Pages: 8583-8589
https://doi.org/10.2298/FIL2325583L
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Quivers associated with finite rings - a cohomological approach
Lipkovski A.T. 
(University of Belgrade, Faculty of Mathematics), aleksandar.lipkovski@matf.bg.ac.rs
Matović J. (née Škorić) (University of Belgrade, Faculty of Mathematics), bubica11@gmail.com
In this paper we present the ongoing research on connection between digraphs
associated to finite (commutative) rings and quiver representations. Digraph
associated to a finite ring A has the set of vertices V = A2 and arrows (or
edges) E = {(x, y) → (x+y, xy), x, y ∈ A}. In another
terminology, it is a finite quiver with loops. In addition to previous work
to understand these graphs, the main goal of the present work is to
introduce some new cohomological and quiver methods. These methods should
provide us with better understanding of properties and classification of
finite rings.
Keywords: Finite commutative rings, Finite digraphs, Homological methods
Show references
A.T. Lipkovski: Digraphs associated with finite rings. Publ de l’Inst Mathématique, Nouvelle série, 92(106) (2012), 35-41.
H. Daoub, O. Shafah, A.T. Lipkovski: An association between digraphs and rings. Filomat 36(3) (2022), 715-720.
A.T. Lipkovski: Structure graphs of rings: definitions and first results. Journal of Mathematical Sciences. 225:4 (2017), 658-665.
I. Dewan: Graph homology and cohomology, preprint (2016).
M. Gavrilović: Graphs associated with rings (Grafovi pridruženi prstenima, in Serbian). Master thesis, University of Belgrade - Faculty of Mathematics (2022), elibrary.matf.bg.ac.rs.
H. Derksen, J. Weyman: Quiver representations. Notices of the AMS, 52:2 (2005), 200-206.
A. Lipkovski, J. Matović (née Škorić): On quiver representations of digraphs associated with finite rings. XXI Geometrical Seminar, Belgrade, June 26 - July 2 (2022), Book of Abstracts p. 37.