Compressed zero-divisor graphs of matrix rings over finite fields

ABSTRACT

We extend the notion of the compressed zero-divisor graph $\Theta \left(R\right)$ to noncommutative rings in a way that still induces a product preserving functor $\Theta$ from the category of finite unital rings to the category of directed graphs. For a finite field F, we investigate the properties of $\Theta \left(M_n\right(F\left)\right)$, the graph of the matrix ring over F, and give a purely graph-theoretic characterization of this graph when $n\ne 3$. For $n\ne 2$ we prove that every graph automorphism of $\Theta \left(M_n\right(F\left)\right)$ is induced by a ring automorphism of $M_n\left(F\right)$. We also show that for finite unital rings R and S, where S is semisimple and has no homomorphic image isomorphic to a field, if $\Theta \left(R\right)\cong \Theta \left(S\right)$, then $R\cong S$. In particular, this holds if $S=M_n\left(F\right)$ with $n\ne 1$.

KEYWORDS:

• Zero-divisor
• compressed zero-divisor graph
• matrix ring
• automorphism

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 16U99
• 15B33
• 05C25
• 05C50

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the Slovenian Research Agency (Javna Agencija za Raziskovalno Dejavnost RS), project number BI-BA/16-17-025.