A test for identities satisfied in lattices of submodules

Suppose R is ring with 1, and HLe(R) denotes the variety of modular lattices generated by the class of lattices of submodules of all R-modules. An algorithm using Mal'cev conditions is given for constructing integers m > 0 and n -> 1 from any given lattice polynomial inclusion formula d ~ e. The main result is that d c e is satisfied in every lattice in H~(R) if and only if there exists x in R such that (m 9 1)x = n 1 in R, where 0-1 = 0 and k 9 1 = 1 + 1 + . . . + 1 (k times) for k => 1. For example, this "divisibility" condition holds for m = 2 and n = 1 if and only if 1 + 1 is an invertible element of R, and it holds for m = 0 and n = 12 if and only if the characteristic of R divides 12. This result leads to a complete classification of the lattice varieties HLe(R), R a ring with 1. A set of representative rings is constructed, such that for each ring R there is a unique representative ring S satisfying HLe(R)= HLe(S). There is exactly one representative ring with characteristic k for each k_-> 1, and there are continuously many representative rings with characteristic zero. If R has nonzero characteristic, then all free lattices in H~(R) have recursively solvable word problems. A necessary and sufficient condition on R is given for all free lattices in ItLe(R) to have recursively solvable word problems, if R is a ring with characteristic zero. All lattice varieties of the form H~(R) are self-dual. A variety H~(R) is a congruence variety, that is, it is generated by the class of congruence lattices of all members of some variety of algebras. A family of continuously many congruence varieties related to the varieties HLa(R) is constructed.