Gabriel-Popescu type theorems and applications

In this paper we obtain a general version of Gabriel-Popescu theorem representing any Grothendieck category as a quotient category of the category of modules over a ring (not necessarily with unit) with enough idempotents to right using a family of generators (U_i)_i∈I of where U_i are not supposed to be small. Applications to locally finite categories are obtained. In particular, for a coalgebra C (over a field) we prove that C is right semiperfect if and only if the category has the AB4∗ condition.