Categorical Quotients

Every equivalence relation R on an algebraic variety U defines a class R of all morphisms constant on equivalence classes of R and determines its categorical closure defined on U by x y if and only if φ(x) = φ(y), for every φ R. It is proved (Theorem A) that the equivalence classes of coincide with fibers of a morphism ψ R. In the family of all morphisms with this property we may determine a subfamily of morphisms, called final pseudoquotients, which contains categorical quotients, if a categorical quotient exists, and, in the general case, is a substitute of such quotients.