Graf Pembagi Nol dari Ring Komutatif ℤp1q1 x ℤp2q2

Abstract

A commutative ring R with zero-divisor Z(R), represented into the zero-divisor graph r(R) whose vertices consist of x,yeZ(R), with distinct vertices x and y adjacent if and only if xy=0. This study analyzed the pattern of the zero-divisor graph of the commutative ring ℤp1q1×ℤp2q2 with 2≤p1,p2 and 2≥q1,q2eℤ+ for any prime p1,p2. Based on the results, the pattern of the zero-divisor graph of r(ℤp2) is a complete graph Kp−1, the pattern of r(ℤp1×ℤp2) is a complete bipartite graph Kp1−1,p2−1, the pattern of r(ℤp1×ℤp22) is a K(p1−1),(p22−1)uK(p2−1),(p1−1)(p2−1) with a connected edge forming a complete graph Kp2−1, and the pattern of r(ℤp12×ℤp22) is a K(p1−1),(p1−1)(p22−1)uK(p12−1),(p22−1)uK(p2−1),(p2−1)(p12−1) with two connected edges, forming complete graphs Kp1−1 and Kp2−1.