Compressed zero-divisor graphs of noncommutative rings

Abstract

We extend the notion of the compressed zero-divisor graph $\Theta (R)$ 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 $\Theta (M_n(F))$, 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 $\Theta (M_n(F))$ is induced by a ring automorphism of $M_n(F)$. 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 $\Theta (R)\cong \Theta (S)$, then $R\cong S$. In particular, this holds if $S=M_n(F)$ with $n\ne 1$.

30 pages

References in corpus (3)

• The total zero-divisor graph of commutative rings
• Compressed zero-divisor graphs of noncommutative rings
• Categorial properties of compressed zero-divisor graphs of finite commutative rings

Cited by in corpus (2)

• Compressed zero-divisor graphs of noncommutative rings
• Compressed commuting graphs of matrix rings