In Commutator Theory, Freese and McKenzie proved that every locally finite abelian Mal'cev variety is finitely based. It was also shown there that there exists to each abelian Mal'cev variety an associated ring and module over said ring. I wanted to see if there was a way to relate the identities of an abelian Mal'cev variety to said ring and module.

Conjecture

The conjecture was that this was possible, and better yet, that finitely presentable ring and module translated to a finitely based variety. Conversely, given a finitely based abelian Mal'cev variety, could one give a finite presentation for the ring and module.

Results

The conjecture turns out to be true, after a lot of technical details relating terms and congruences of the free algebras to the the terms and congruences of the associated ring and module. One immediate application is on the varieties of R-modules for a given ring R. Namely, a variety of R-modules is finitely based if and only if R is finitely presentable.

Future Directions

The abelian assumption gives plenty of structure. In Commutator Theory, there is an easy to read basis of abelian algebras in a Mal'cev variety. The book also shows how abelian varieties have an abelian group structure on terms. The basis for k-nilpotent algebras grows exponentially and becomes cumbersome to read, let alone work with. Also, the group structure is now k-nilpotent instead of abelian. This removes any chance of finding a ring and module structure on k-nilpotent Mal'cev varieties. I still think that one can find a general result relating finitely based k-nilpotent Mal'cev varieties to some other finitely presentable algebras.