Journals / Turkish Journal of Mathematics / 2013 / Cilt: 37 - Sayı: 4
On the nilpotent graph of a ring
- Pages
- 553–559
- DOI
- —
Abstract
Let R be a ring with unity. The nilpotent graph of R, denoted by GN(R), is a graph with vertex set ZN(R)* = {0 \neq x \in R \mid xy \in N(R) for some 0 \neq y \in R}; and two distinct vertices x and y are adjacent if and only if xy \in N(R), where N(R) is the set of all nilpotent elements of R. Recently, it has been proved that if R is a left Artinian ring, then diam(GN(R)) \leq 3. In this paper, we present a new proof for the above result, where R is a finite ring. We study the diameter and the girth of matrix algebras. We prove that if F is a field and n \geq 3, then diam(GN(Mn(F))) = 2. Also, we determine diam (GN (M2(F))) and classify all finite rings whose nilpotent graphs have diameter at most 3. Finally, we determine the girth of the nilpotent graph of matrix algebras.
Özet
Let R be a ring with unity. The nilpotent graph of R, denoted by GN(R), is a graph with vertex set ZN(R)* = {0 \neq x \in R \mid xy \in N(R) for some 0 \neq y \in R}; and two distinct vertices x and y are adjacent if and only if xy \in N(R), where N(R) is the set of all nilpotent elements of R. Recently, it has been proved that if R is a left Artinian ring, then diam(GN(R)) \leq 3. In this paper, we present a new proof for the above result, where R is a finite ring. We study the diameter and the girth of matrix algebras. We prove that if F is a field and n \geq 3, then diam(GN(Mn(F))) = 2. Also, we determine diam (GN (M2(F))) and classify all finite rings whose nilpotent graphs have diameter at most 3. Finally, we determine the girth of the nilpotent graph of matrix algebras.