2013/05/27 by M. J. Nikmehr, Nikmehr, M. J., F. Heydari +1
Mathematics · #05C25 #05C69 #13A #13C99 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #math.CO #msc:05C25 #msc:05C69 #msc:13A #msc:13C99
paper · pdf · doi:10.48550/arxiv.1305.6199
This paper has been withdrawn by the author because of some typos and errors in the proofs
openalex publication_date 2013/05/27 · arxiv created 2013/07/28 · arxiv updated 2013/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a commutative ring and M be an R-module, and let Z(M) be the set of all zero-divisors on M. In 2008, D.F. Anderson and A. Badawi introduced the regular graph of R. In this paper, we generalize the regular graph of R to the M-regular graph of R, denoted by M-Reg(Γ(R)). It is the undirected graph with all M-regular elements of R as vertices, and two distinct vertices x and y are adjacent if and only if x+y∈ Z(M). The basic properties and possible structures of the M-Reg(Γ(R)) are studied. We determine the girth of the M-regular graph of R. Also, we provide some lower bounds for the independence number and the clique number of the M-Reg(Γ(R)). Among other results, we prove that for every Noetherian ring R and every finitely generated module M over R, if 2∉ Z(M) and the independence number of the M-Reg(Γ(R)) is finite, then R is finite.