Author:
,Ali Asma,Ahmad Bakhtiyar,
Abstract
Let A be a commutative ring with non-zero identity, and E(A)={p∈A|annA(pq)≤eA,for some q∈A∗}.The extended essential graph, denoted by EgG(A) is a graph with the vertex set E(A)∗=E(A)\{0}. Two distinct vertices r,s∈E(A)∗ are adjacent if and only if annA(rs)≤eA. In this paper, we prove that EgG(A) is connected with diam(EgG(A))≤3 and if EgG(A) has a cycle, the ngr(EgG(A))≤4. Furthermore, we establishthat if A is an Artinian commutative ring, then ω(EgG(A))=χ(EgG(A))=|N(A)∗|+|Max(A)|.
Publisher
Luhansk Taras Shevchenko National University
Reference13 articles.
1. [1] Akbari, S., Mohammadian, A.: On the zero-divisor graph of a commutative ring.J. Algebra.274, 847-855 (2004). https://doi.org/10.1016/S0021-8693(03)00435-6
2. [2] Anderson, D.D., Naseer, M.: Beck's coloring of a commutative ring. J. Algebra.159, 500-514 (1993). https://doi.org/10.1006/jabr.1993.1171
3. [3] Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring.J. Algebra.217, 434-447 (1999). https://doi.org/10.1006/jabr.1998.7840
4. [4] Anderson, D.F., Levy, R., Shapirob, J.: Zero-divisor graphs, von Neumann regu-lar rings and Boolean algebras. J. Pure Appl. Algebra.180, 221-241 (2003). https://doi.org/10.1016/S0022-4049(02)00250-5
5. [5] Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Massachusetts, London, Ontario (1969).