Biography

Ayman Badawi

American University of Sharjah, United Arab Emirates

**Title:** On the Annihilator Graph of A Commutative Ring

Let R be a commutative ring with nonzero identity, Z(R) be its set of zero- divisors, and if a 2 Z(R), then let annR(a) = fd 2 R j da = 0g. The annihilator graph of R is the(undirected) graph AG(R) with vertices Z(R) = Z(R) n f0g, and two distinct vertices x and y are adjacent if and only if annR(xy) 6= annR(x) [ annR(y). It follows that each edge (path) of the zero-divisor graph ??(R) is an edge (path) of AG(R). In this paper, we study the graph AG(R). For a commutative ring R, we show that AG(R) is connected with diameter at most two and with girth at most four provided that AG(R) has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG(R) is identical to the zero-divisor graph ??(R) if and only if R has exactly two minimal prime ideals.

**References **

[1] D. F. Anderson, On the diameter and girth of a zero-divisor graph, II, Houston J. Math. 34(2008), 361{371.

[2] D.F. Anderson and S. B. Mulay, S.B., On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2) (2007), 543-550.

[3] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217(1999), 434{447.

[4] I. Beck, Coloring of commutative rings, J. Algebra 116(1988), 208-226.

[5] T. G. Lucas, The diameter of a zero-divisor graph, J. Algebra 301(2006), 3533{3558.

Ayman Badawi is a full Professor at the Department of Mathematical and Statistical Sciences at the American University of Sharjah. He holds a Ph.D. in Mathematics from the University of North Texas. His research interests are in the area of Commutative Ring Theory and Graphs associated to Rings. Dr. Badawi has received local and international recognition in his field of research, and is the editor in chief of the Palestine Journal of Mathematics (PJM), on the editorial board of the scientific world Journal (Algebra section) and AnaleleStiintifice ale UniversitatiiOvidius Constanta. He has numerous publications, including book chapters, journal articles, and conference proceedings. A complete list of his works can be found on his university webpage : https://www2.aus.edu/facultybios/profile.php?faculty=abadawi and hispersonal page: www.ayman-badawi.com

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