Firm homomorphisms of rings and semigroups

In this paper we define firm homomorphisms between rings without identity in such a way that the category of rings with identity will become a full subcategory of the category of firm rings with firm homomorphisms as morphisms. We prove that firm homomorphisms are in one-to-one correspondence with pairs of compatible concrete functors between certain module categories. This correspondence is given by the restriction of scalars. We also prove the semigroup theoretic analogues of these results and give a list of examples of firm homomorphisms.