# When the annihilator graph of a commutative ring is planar or toroidal?

## DOI:

https://doi.org/10.12697/ACUTM.2020.24.19## Keywords:

Annihilator graph, planarity, toroidality## Abstract

Let *R* be a commutative ring with identity, and let *Z(R)* be the set of zero-divisors of *R*. The annihilator graph of *R* is defined as the undirected graph *AG(R)* with the vertex set *Z(R)* = Z(R) \ {0}*, and two distinct vertices* x* and *y* are adjacent if and only if ann_R(xy) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified.

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