Varieties of Associative Rings Containing a Finite Ring that is Nonrepresentable by a Matrix Ring Over a Commutative Ring

Abstract

In this paper, we give examples of infinite series of finite rings B_v^(m) , where m ≥ 2, 0 ≤ v ≤ 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 var B_v^(m) , where m = 2 or m − 1 = (p − 1)k, k ≥ 1, and p ≥ 3 or p = 2 and m ≥ 3, 0 ≤ 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.