It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g”. In this note we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham’s theorem to arbitrary (not necessary finite) commutative rings.

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[12] C. Wickham, Rings whose zero-divisor graphs have positive genus, J. Algebra 321
(2009) 377-383.