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 $\operatorname{Th}(R)$ 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 $R$?
Bibliography: 20 titles.