Homomorphisms

A homomorphism is a structure-preserving map between two algebraic structures of the same kind. It respects the defining operation(s) of the structures, so the image of a product or sum under the map corresponds to the product or sum of the images. For example, if f: G -> H is a group homomorphism, then f(xy) = f(x)f(y) for all x and y in G. In rings, a homomorphism f satisfies f(x + y) = f(x) + f(y) and f(xy) = f(x)f(y). For vector spaces and modules, homomorphisms are linear maps: f(a x + b y) = a f(x) + b f(y).

Different algebraic contexts use slightly different conventions about preserving identities. Ring homomorphisms are sometimes required to map the multiplicative identity to the multiplicative identity; when this is not imposed, they may be called non-unital ring homomorphisms. In more general settings, homomorphisms are defined to preserve all relevant operations of the structure.

Key concepts associated with homomorphisms include the kernel and the image. The kernel of f is the set of elements that map to the identity of the target, and in groups or rings it is a normal subgroup or an ideal, respectively. The image is the set of all outputs f(A). The First Isomorphism Theorem states that the quotient of the domain by the kernel is isomorphic to the image, encapsulating how information is lost and preserved by f.

Other terms arise from the same idea: endomorphisms are homomorphisms from a structure to itself, and automorphisms are isomorphisms from a structure to itself. Homomorphisms are the morphisms in the category-theoretic sense and serve as the basic links between algebraic objects. Examples include the reduction map Z -> Z/nZ and linear transformations between vector spaces.