Some aspects of zero-divisor graphs for the ring of Gaussian integers modulo $2^{n}$

Abstract

For a commutative ring R with unity ($1\ne 0$), the zero-divisor graph of R is a simple graph with vertices as elements of $Z(R)^{*}=Z(R)\setminus \{ 0 \}$, where Z(R) is the set of zero-divisors of R and two distinct vertices are adjacent whenever their product is zero. An algorithm is presented to create a zero-divisor graph for the ring of Gaussian integers modulo $2^{n}$ for $n\ge 1$. The zero-divisor graph $\Gamma (\mathbb {Z}_{2^{n}}[i])$ can be expressed as a generalized join graph $G[G_{1}, \dots , G_{j}]$, where $G_{i}$ is either a complete graph (including loops) or its complement and G is the compressed zero-divisor graph of $\mathbb {Z}_{2^{2n}}$. Next, we show that the number of isomorphisms between the zero-divisor graphs for the ring of Gaussian integers modulo $2^{n}$ and the ring of integers modulo $2^{2n}$ is equal to $\prod _{j=1}^{2(n-1)}2^{j}!$.