On the Ideal Based Zero Divisor Graphs of Unital Commutative Rings and Galois Ring Module Idealizations

Let R be a commutative ring with identity 1 and I is an ideal of R. The zero divisor graph of the ring with respect to ideal has vertices defined as follows: {u ∈ Ic | uv ∈ I for some v ∈ Ic}, where Ic is the complement of I and two distinct vertices are adjacent if and only if their product lies in the ideal. In this note, we investigate the conditions under which the zero divisor graph of the ring with respect to the ideal coincides with the zero divisor graph of the ring modulo the ideal. We also consider a case of Galois ring module idealization and investigate its ideal based zero divisor graph.