ON A ZERO-DIVISOR BASED GRAPH STRUCTURE OF A COMMUTATIVE RING
PRIYANKA PRATIM BARUAH *
Department of Mathematics, Girijananda Chowdhury Institute of Management and Technology, Guwahati – 781017, India.
KUNTALA PATRA
Department of Mathematics, Gauhati University, Guwahati – 781014, India.
*Author to whom correspondence should be addressed.
Abstract
The concept of a zero-divisor based graph structure of a commutative ring R is introduced in this paper and the graph is denoted by ΓZ (R). In our discussion, we study the ring-theoretic properties of R and the graph-theoretic properties of ΓZ(R). In our investigation, we characterize some basic properties of ΓZ (R) related to connectedness, diameter and girth. We show that ΓZ (R) is connected and the diameter of ΓZ (R) is at most 2. If ΓZ (R) contains a cycle, then we show the girth of ΓZ (R) is at most 3. We examine the diameter of ΓZ (R) for a direct product R = R1 x R2 of two commutative rings R1 and R2 with respect to the zero-divisors and regular elements of R1 and R2. Then we study the diameter of ΓZ (R) for a finite direct product R = R1 x R2 x R3 x…x Rn (n > 2) of commutative rings R1, R2, R3,…,Rn (n > 2) with respect to the zero-divisors and regular elements of R1, R2, R3,…,Rn (n > 2).
Keywords: Zero-divisor, commutative ring, diameter, girth, finite direct product