Speaker
Description
In this talk we demonstrate how a matrix algebra over a finite field can be completely described using combinatorial properties. The main tool that allows one to do this is the compressed zero-divisor graph of a ring, which describes pairs of matrices $A$ and $B$ such that $AB=0$. We list a set of $5$ combinatorial axioms that uniquely determine the compressed zero-divisor graph $\Theta(M_n(\mathbb{F}))$ of the matrix algebra $M_n(\mathbb{F})$, where $|\mathbb{F}|=p^m$. Furthermore, the structure of this graph uniquely determines the ring $M_n(\mathbb{F})$ itself up to isomorphism. We also discuss some properties of the compressed zero-divisor graphs that may be useful for investigating subalgebras of $M_n(\mathbb{F})$ and individual matrices.