The isomorphism theorems hold for module homomorphisms. Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms). The composition of two ring homomorphisms is a ring homomorphism. The canonical homomorphisms from "R" to the quotients formula 204 induce a homomorphism formula 205. Functors between one-object categories correspond to monoid homomorphisms. For a detailed discussion of relational homomorphisms and isomorphisms see. Module homomorphisms between finitely generated free modules may be represented by matrices. The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms. Consider the category formula 141 of abelian groups and group homomorphisms. Functors and natural transformations abound in algebraic topology, with the Hurewicz homomorphisms serving as examples. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon. Conditions characterizing monoid homomorphisms are discussed further. Then any finite family of distinct monoid homomorphisms is linearly independent. In algebra, epimorphisms are often defined as surjective homomorphisms. Commutative rings, together with ring homomorphisms, form a category.