Logic in representations of groups

Let $K$ be a commutative and associative ring with unit. We consider representations of groups over $K$ from the viewpoint of some logic. In particular, different logical invariants of representations, as well as relations between the various representations corresponding to the invariants, are studied. One of the basic relations is isotypeness. Here we use the concept of a type adopted in model theory. This paper adjoins results by B. I. Plotkin in [Isotyped algebras, in Mal'tsev Readings, Sovr. Mat. Pril., 15, MIAN, Moscow, 40–66 (2011)], where similar results were derived for one-sorted algebras.