Directed prime graph of non-commutative ring

Document Type : Research Paper


1 Department of Mathematics, Gauhati University, Guwahati- 781014, Assam, India

2 Department of Mathematics, Gauhati University, Guwahati- 781014, Assam, India



Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the paper.


[1] S. Akbari, A. Mohammadian , On The Zero-Divisor Graph of A Commutative Ring, J. Algebra , Vol 274 (2004), 847-855.
[2] S. Akbari, A. Mohammadian, Zero-Divisor Graphs of Non-Commutative Rings, J. Algebra, Vol 296 (2006), 462-479.
[3] D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring, J.Algebra 217 (1999), No. 2, 434 - 447.
[4] D. F. Anderson, S. B. Mulay, The Diameter and Girth of A Zero-Divisor Graph, J. Pure App. Algebra, Vol 210(2007), No 2, 543550 .
[5] S. E. Atani, A. Y. Darani , Zero-Divisor Graphs with respect to Ideals in Non-commutative Rings, ISRN Discrete Math., Vol (2011) (Article ID 459547, 7 pages, doi:10.5402/2011/459547)
[6] I. Beck , Coloring of commutative rings, J. Algebra 116 (1988) No. 1, 208 - 226.
[7] I. Bozic, Z. Petrovic, Zero-Divisor Graphs Of Matrices Over Commutative Rings, Comm. Alg. Vol 37 (2009), 1186 -1192.
[8] F. Harary, Graph Theory, Eddison Wesley Publishing Company inc. 1969.
[9] J. Lambek, Lectures on Rings and Modules , Blaisdel Publ. Co., 1966.
[10] T. G. Lucas , The Diameter Of A Zero Divisor Graph, J. Algebra, Vol 301(2006), 174193.
[11] H. R. Maimani, M. R. Pournaki, S. Yessemi, Zero Divisor Graph with respect to An Ideal, Comm. Alg., Vol. 34(2006), 923929.
[12] K. K. Rajkhowa, H. K. Saikia , Zero-Divisor Graphs Of Non-Commutative Rings, Adv. Appl. Discrete Math., Vol. 10 (2012), No. 1, 49- 64.
[13] S. P. Redmond, The Zero-Divisor Graph Of A Non-Commutative Ring, International J. Commutative Rings, Vol 1(4) (2002), 203 211.
[14] Bh. Satyanarayana, K. Shyam Prasad, D. Nagaraju, Prime Graph of a Ring, J. of Combinatorics, Information and System Sources, Vol 35 (2010) No 1-2, 27-42.