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.