k field
A finite dimensional k-algebra (associative, with 1, usually not semisimple).
n number of simple (= irreducible) A-modules.
Modules are usually finitely generated left A-modules
mod A the category of these A-modules
(a length category: abelian, all objects have finite length)
Two algebras A, B are called Morita equivalent,
provided the categories mod A and mod B are equivalent.
thus provided there exists a progenerator in mod A with
endomorphism ring B
(P progenerator iff P is finitely generated, projective, and a generator)
If P is a progenerator with B = End P,
then Hom(P, - ) : mod A → mod B is an equivalence