Zero-divisor graphs of genus one
The goal of the present paper is to describe all rings of two special types which have genus at most one. The first type studied is represented by finite direct products of \(\mathbb{Z}_{p_i^{n_i}}\) (where all \(p_i\) are primes), while the second by the factor rings \(\mathbb{Z}_n[x]/(x^m)\) (where \(m\) and \(n\) are at least \(2\)). Moreover, Theorem 1.4 of \textit{S. Akbari, H. R. Maimani} and \textit{S. Yassemi} [J. Algebra 270, No. 1, 169--180 (2003; Zbl 1032.13014)] is generalized (see Theorem 3.6) and there is given an affirmative answer to a question raised in [loc. cit.] (``Is it true that, for any local ring \(R\) of cardinality 32, which is not a field, the graph \(U_{393}(R)\) introduced by \textit{D. F. Anderson} and \textit{P. Livingston} [J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062)] is not planar?'').
scientific article; zbMATH DE number 5077817