ideals in additive categories

Let C be an additive category. For every pair X,Y of objects of C, let A(X,Y) be a subgroup of the additive group Hom(X,Y). Assume that for every quadruple W,X,Y,Z of objects of C, we have Hom(W,X)xA(X,Y)xHom(Y,Z) --> Hom(W,Z) (by composition ) maps into A(W,Z). That seems like having an ideal and it seems like I should then be able to form a category in which the set of morphisms from an object X to an object Y is Hom(X,Y)/A(X,Y). Is there a name for such a system A and is there a name for the construction of modding out by A? Where is this formalized? Allan Adler ara@altdorf.ai.mit.edu ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++