Quillenʹs small object argument in the category of firm modules

Firm modules over a nonunital ring are introduced by Quillen to study properties of Morita invariance extending thus the case of unitary rings. In the present paper we introduce new homology groups in the category of firm modules which are preserved under Morita equivalences in very general terms. We see that there are long exact sequences of homology attached to certain morphisms of firm modules. Our arguments are based in Quillenʹs small object argument.