Non-Jordaness of the automorphism group of the zero-divisor graph of a matrix ring over number rings

Abstract

We provide a construction of the induced subgraphs of the zero-divisor graph of $M_2(R)$ for the ring $R$ of algebraic integers of some number fields that are neither complete nor connected, and study the structure of the induced subgraphs explicitly. As an application, we prove that the automorphism group of the zero-divisor graph of $M_2(R)$ is not a Jordan group.

The title of the paper is changed, and the affiliation of the second author is updated. Any valuable comments are welcome