Recovering rings from zero-divisor graphs (Q2847692)

For a commutative ring \(R\) with \(1\), the zero-divisor graph of \(R\), denoted \(\Gamma(R)\), is the simple graph with vertex set equal to the nonzero zero-divisors of \(R\) and with distinct vertices \(x\) and \(y\) adjacent if and only if \(xy = 0\).NEWLINENEWLINEIn the paper under review the author shows that given a zero-divisor graph \(G\) with four or more vertices, one can find a ring \(R\cong R_1\times R_2\times\cdots\times R_n\), where each \(R_i\) is local such that the zero divisor graph of \(R\) is \(G\) and such that this representation is unique up to factors \(R_i\) of the same zero-divisor class. Furthermore, corollaries to this result prove that when the zero-divisor graphs have four or more vertices, local rings and non-local rings cannot have isomorphic zero-divisor graphs.