Some Properties of the Complement of the Zero-Divisor Graph of a Commutative Ring

Abstract

Let R be a commutative ring with identity admitting at least two nonzero zero-divisors. Let Γ R c denote the complement of the zero-divisor graph Γ R of R. It is shown that if Γ R c is connected, then its radius is equal to 2 and we also determine the center of Γ R c . It is proved that if Γ R c is connected, then its girth is equal to 3, and we also discuss about its girth in the case when Γ R c is not connected. We discuss about the cliques in Γ R c .

Complement of the Zero Divisor Graph of A lattice

“…The investigation of graphs associated with various algebraic structures is a very important topic, well known and established in modern algebra. The graphs have played crucial roles in the study of ring constructions and their applications in coding theory (see Alfaro and Kelarev [2,3] and Bereg et al [6]) and automata theory (see [16][17][18]), and in the study of commutative rings and semigroups (see [1,4,5,19,22,23]).…”

When is the complement of the zero-divisor graph of a commutative ring planar?

“…Let R be a commutative ring with identity which has exactly three maximal N -primes of (0). Let {P 1 , P 2 , P 3 } be the set of all maximal N -primes of (0) in R. The following statements are equivalent: [17,Proposition 4.4] that R is finite. Hence, each prime ideal of R is a maximal ideal of R. Now we obtain from the Chinese remainder theorem that R = R/(0) = R/ ∩ 3 i=1 P i ∼ = R/P 1 × R/P 2 × R/P 3 as rings.…”

“…Then ω((Γ(R)) c ) ≤ 5. Hence, it follows from [17,Proposition 4.4] that R is finite. Note that (ii) follows as in the proof of (i) ⇒ (ii) with the help of Lemmas 2.2 and 2.3.…”

“…Hence ω((Γ(R)) c ) ≤ 5. Therefore, we obtain from [17,Proposition 4.4] that R is finite. Since any prime ideal of a finite ring is maximal, it follows from [14,Theorem 84] that any prime ideal of R is a subset of the set of zero-divisors of R. Hence, it follows that {P 1 , P 2 } is the set of all prime ideals of R. Let (0) = q 1 ∩ q 2 be the irredundant primary decomposition of (0) in R where q i is P i -primary for i = 1, 2.…”

“…Let P be the unique maximal N -prime of (0) in T . It was shown in [17,Proposition 4.21] that if ω((Γ(T )) c ) < ∞, then P is nilpotent. Indeed, it was shown in [17,Proposition 4.21] that P n = (0) where n = (ω((Γ(T )) c )) 2 + 1.…”