Akademik Veri Yönetim Sistemi
COMMUNICATIONS IN ALGEBRA, cilt.2025, ss.1-16, 2025 (SCI-Expanded, Scopus)
Let R be a commutative ring with1 ̸= 0 and W∗(R) its set of nonzero nonunits.
In this paper, we introduce and study a new graph that is closely related to
the cozero-divisor graph ′(R). We define the extended cozero-divisor graph of
R to be the (simple) graph ′(R) with vertices W∗(R), and distinct vertices x
and y are adjacent if and only if there are positive integers m and n such that
x^m / ∈ y^nR and y^n / ∈ x^mR. We determine when ′(R)= ′(R) and show that
′(R) is a complete graph if and only if aR = {0,a} for every a ∈ W∗(R). We
also study the diameter and girth of ′(R) and show that gr( ′(R1 × R2))= 3
if and only if either R1 is not a valuation ring, R2 is not a valuation ring, or R1
and R2 are not fields, and that gr(′(R))= gr( ′(R)) ∈ {3,4,∞} when R is a
commutative Artinian ring. Many examples are given to illustrate the theory
along with some new results for ′(R).