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Erschienen in:
01.03.2021 | Original Research
Some aspects of zero-divisor graphs for the ring of Gaussian integers modulo \(2^{n}\)
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1/2022
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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 \(\prod _{j=1}^{2(n-1)}2^{j}!\).