Locally finite ring varieties

Abstract

Necessary and sufficient conditions are given for a variety of associative rings to be locally finite. These conditions are utilized to show that a variety is generated by a finite ring if, and only if, it contains only finitely many subvarieties. Also, the Everett extension of a variety by another variety is a locally finite variety (a variety generated by a finite ring) if, and only if, each of the varieties is locally finite (generated by a finite ring). All rings considered here are associative and not necessarily with 1. A variety (an equational class) of algebras is a class of algebras closed under homomorphic images, subalgebras and Cartesian products; or equivalently, it is the class of all algebras satisfying a set of identities (cf., e.g., [1], [2], [3], [9], [10], [11]). A variety is said to be locally finite if every finitely generated member is finite. We show that a locally finite ring variety is precisely a variety satisfying mx = 0 and x' + xr+lq(x) = 0 for some positive integers m, r and some q(x) c Z[x]. In [6], R. L. Kruse shows that the identities of a finite-ring are finitely based; the arithmetical ring case is shown by H. Werner and R. Wille [13]. From [6], it also follows that a finite ring generates a variety containing only finitely many subvarieties. We show here that the converse is also true. If U and 13 are ring varieties, then the class U . 13 of all rings possessing an ideal belonging to 11 whose factor belongs to E3 is a variety [4], [8], [9], [10]. We show that the set of all locally finite varieties (varieties generated by a finite ring) is closed under products and hence under lattice joins and meets. 1. In a locally finite variety, every member generated by one element is finite. There are varieties of groups whose cyclic members are finite, but the varieties are not locally finite: the Burnside varieties of groups, x = 1, n is sufficiently large [10]. The situation is different for rings. Presented to the Society, January 23, 1975; received by the editors March 21, 1974. AMS (MOS) subject classifications (1970). Primary 08A15, 16A38; Secondary 16A06, 16A44.