Existence varieties of regular rings and complemented modular lattices

Goodearl, Menal, and Moncasi [K.R. Goodearl, P. Menal, J. Moncasi, Free and residually artinian regular rings, J. Algebra 156 (1993) 407–432] have shown that free regular rings with unit are residually artinian. We extend this result to the case without unit and use it to derive that free regular rings as well as free complemented (sectionally complemented) Arguesian lattices are residually finite. Here, quasi-inversion for rings and complementation (sectional complementation, respectively) for lattices are considered as fundamental operations in the appropriate signature. It follows that the equational theory of each of the classes listed above is decidable. The approach is via so-called existence varieties in ring or lattice signature. Those are classes closed under operators H, S, and P within the class of all regular rings or the class of all sectionally complemented modular lattices. We show that any existence variety in the considered classes is generated by its artinian or finite height members.