Varieties of associative rings containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring

In this paper, we give examples of infinite series of finite rings $B_v^{(m)}$, where $m\geq2$, $0\leq v\leq p-1$, and $p$ is a prime number, that are not representable by matrix rings over commutative rings, and we describe the basis of polynomial identities of these rings. We prove here that every variety $\operatorname{var}B_v^{(m)}$, where $m=2$, or $m-1=(p-1)k$, $k\geq1$, and $p\geq3$, or $p=2$, $m\geq3$, $0\leq v<p$, and $p$ is a prime number, is a minimal variety containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring. Therefore, we describe almost finitely representable varieties of rings whose generating ring contains an idempotent element of additive order $p$.