SOME RESULTS ON COZERO-DIVISOR GRAPH OF A COMMUTATIVE RING

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Gamma'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a is not an element of Rb and b is not an element of Ra. In this paper, we show that if Gamma'(R) is a forest, then Gamma'(R) is a union of isolated vertices or a star. Also, we prove that if Gamma'(R) is a forest with at least one edge, then R congruent to Z(2) circle plus F, where F is a field. Among other results, it is shown that for every commutative ring R, diam(Gamma(R[x])) = 2. We prove that if R is a field, then Gamma'(R[[x]]) is totally disconnected. Also, we prove that if (R, m) is a commutative local ring and m not equal 0, then diam(Gamma (R[[x]])) <= 3. Finally, it is proved that if R is a commutative non-local ring, then diam(Gamma'(R[[x]])) <= 3.