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 Bv (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 Bv (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. © 2016, Springer Science+Business Media New York.