The author characterizes a cocomplete abelian category C by a subfunctor F of the functor \(Hom_ C(U\),-) for suitable \(U\in C\). If F is faithful, exact, and preserves coproducts (the pair (U,F) is called a subprogenerator in this case), then C is equivalent to the category of s- unital right modules over a right s-unital ring. One of the consequences is a Freyd-Gabriel theorem on equivalence of module categories.
Several important results on s-unital modules and rings are stated. Examined is also a subprogenerator of a closed subcategory of a module category with a generalization of Fuller’s theorem on equivalences. Some results on Morita equivalence are generalized by considering a subprogenerator of the category of s-unital right modules over a right s- unital ring. After defining quotient rings of right s-unital rings, the author proves that quotient rings of right Morita equivalent right s- unital rings are also right Morita equivalent.