Algebra Universalis, 23 (1986) 3 2 - 4 3 0002-5240/86/010032-12501.50 + 0.20/0 9 1986 Birkh~userVerlag, Basel Identities in lattices of ring varieties M. V. VOLKOV To the memory of Andrds Huhn 1. Introduction and main results In [1], the author had introduced the notion of a joined variety of rings. The importance of this notion may be explained by the following. In the class of joined varieties, it is possible to prove, for a large spectrum of lattice properties zL reduction theorems of the following kind: "for a variety Y. if a property A holds in the lattice of subvarieties of the variety Y ~ ~,, for every n, where ~n is the variety of all rings satisfying the identity nx = 0, then A holds in the lattice of subvarieties of Y". (See for example theorem 3 and corollaries 9-11 in [1].) In particular, distributivity was one of the properties considered in [11. A natural question arises about the existence of analogous theorems for o~her lattice properties expressed as identities. The aim of the present work is to give a complete answer to this question. We give necessary notations and definitions. Let ~.)~and ~3~be varieties of (not necessarily associative) rings. The join of ~ and 9~. i.e. the smallest variety containing both ~ and ~, is denoted by ~)~ v ~J~. The set of all subvarieties of a given variety Y forms a lattice L(Y) with the operations of taking join and intersection of varieties. The variety generated by a ring R will be denoted by var (R). The free ring of countable rank in a variety .~ is denoted by F(Y). If A is an abelian group, then T(A) will be its maximal periodic subgroup. A variety Y is called joined if for every subvariety N c_ ,t~ there exists a positive integer n = n(~l) such that nT(F(~))= 0. (This definition is equivalent to the definition given in [1], by theorem 2 from [1].) THEOREM 1. Let E be any variety of lattices and Y be a joined variety of rings. If the lattice L(Y N 23n) belongs to ~ for every n, then the lattice L(Y) belongs to ~.. Presented by Ralph McKenzie~Received January 6, 1983. Accepted for publication in final form August 18, 1985. 32