Counting core sets in matrix rings over finite fields

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