The structure of models and a decidability criterion for complete theories of finite-dimensional algebras

The author gives an algebraic description of models of the theory Th(R) and a criterion for its decidability for an arbitrary (not necessarily associative, and possibly without an identity) finite-dimensional k-algebra over a field k of arbitrary characteristic. This description is based on the solution of the following purely algebraic problem: how completely can the structure of a k-module R be recovered, knowing only the ring operations of R? 1990 American Mathematical Socicty.