2025/09/15 by Djuang, Felicia Servina, Wijayanti, Indah Emilia, Susanti, Yeni
#05C25 #05C60 #13A70 #16U40 #16U60 #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2509.12004
Let R be a finite ring with identity. The clean graph Cl(R) of a ring R is a graph whose vertices are pairs (e, u), where e is an idempotent element and u is a unit of R. Two distinct vertices (e, u) and (f, v) are adjacent if and only if ef = fe = 0 or uv = vu = 1. The graph Cl2(R) is the induced subgraph of Cl(R) induced by the set \(e, u): e is a nonzero idempotent and u is a unit of R\. In this study, we present properties that arise from the isomorphism of two clean graphs and conditions under which two clean graphs over direct product rings are isomorphic. We also examine the structure of the clean graph over the ring M2(ℤp) through their Cl2 graph.