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On the nilpotent graph of a ring
- Source :
- Volume: 37, Issue: 4 553-559, Turkish Journal of Mathematics
- Publication Year :
- 2014
- Publisher :
- TÜBİTAK, 2014.
-
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.
Details
- Language :
- Turkish
- ISSN :
- 13000098 and 13036149
- Database :
- OpenAIRE
- Journal :
- Volume: 37, Issue: 4 553-559, Turkish Journal of Mathematics
- Accession number :
- edsair.doi.dedup.....a35cdcabbd2c2a84a66183118fda38e3