In 1955, Szmielew proved that the theory of abelian groups, equivalently
the theory of Z-modules, is decidable. On the other hand, in the mid
70's Baur and independently Kokorin and Mart'yanov proved that the
theory of k<x,y>-modules is undecidable for any field k.
In this talk, I will give an overview of decidability and undecidability
results for theories of modules.
I will present modern techniques for proving decidability of theories of
I will then specialise to recent results for finite-dimensional
algebras, commutative domains and if time permits, finite commutative rings.