Counting core sets in matrix rings over finite fields

Abstract

Let $R$ be a commutative ring and $M_n(R)$ be the ring of $n\times n$ matrices with entries from $R$. For each $S\subseteq M_n(R)$, we consider its (generalized) null ideal $N(S)$, which is the set of all polynomials $f$ with coefficients from $M_n(R)$ with the property that $f(A)=0$ for all $A\in S$. The set $S$ is said to be core if $N(S)$ is a two-sided ideal of $M_n(R)[x]$. It is not known how common core sets are among all subsets of $M_n(R)$. We study this problem for $2\times 2$ matrices over ${\mathbb{F}}_q$, where ${\mathbb{F}}_q$ is the finite field with $q$ elements. We provide exact counts for the number of core subsets of each similarity class of $M_2({\mathbb{F}}_q)$. While not every subset of $M_2({\mathbb{F}}_q)$ is core, we prove that as $q\to \infty$, the probability that a subset of $M_2({\mathbb{F}}_q)$ is core approaches 1. Thus, asymptotically in~$q$, almost all subsets of $M_2({\mathbb{F}}_q)$ are core.