Homomorphism
A homomorphism is a map between algebraic structures that preserves the operations defining those structures. It expresses a correspondence in which algebraic combinations formed before applying the map agree with the corresponding combinations formed after applying it. Homomorphisms therefore provide the standard mathematical language for comparing structures without requiring a reversible correspondence between their elements.
For structures (A) and (B) of the same algebraic type, a function
[ f\colon A\to B ]
is a homomorphism when every basic operation is preserved. If the type contains an (n)-ary operation (\omega), the defining condition is
[ f\bigl(\omega^A(a_1,\ldots,a_n)\bigr)
\omega^B\bigl(f(a_1),\ldots,f(a_n)\bigr) ]
for all (a_1,\ldots,a_n\in A). Nullary operations, which represent distinguished constants, are preserved by the corresponding condition (f(c^A)=c^B). This formulation belongs to universal algebra, where groups, rings, modules, lattices, and related systems are treated through a common language of operations and identities.
Algebraic meaning
A homomorphism retains equations constructed from the designated operations. If two algebraic expressions have the same value in the domain, their images under a homomorphism also have the same value in the codomain. The converse need not hold, because distinct domain elements can have the same image.
This possible identification of elements distinguishes a homomorphism from an isomorphism. An isomorphism is a bijective homomorphism whose inverse is also a homomorphism. For algebraic structures defined entirely by operations, every bijective homomorphism has a homomorphic inverse. A homomorphism from a structure to itself is an endomorphism, while an invertible endomorphism is an automorphism.
Identity maps are homomorphisms, and the composite of two homomorphisms is again a homomorphism. Algebraic structures of a fixed type consequently form a category whose morphisms are the appropriate homomorphisms.
Principal forms
Group homomorphisms
For groups (G) and (H), a function (f\colon G\to H) is a group homomorphism when
[ f(xy)=f(x)f(y) ]
for every (x,y\in G). Preservation of the identity element and inverses follows from this condition:
[ f(e_G)=e_H, \qquad f(x^{-1})=f(x)^{-1}. ]
The kernel is the normal subgroup
[ \ker f={x\in G:f(x)=e_H}, ]
and the image is the subgroup
[ \operatorname{im}f={f(x):x\in G}. ]
The kernel records precisely which elements become indistinguishable from the identity under the map. More generally, two elements (x) and (y) have the same image exactly when (x^{-1}y\in\ker f).
Ring homomorphisms
A homomorphism (f\colon R\to S) between rings preserves addition and multiplication:
[ f(x+y)=f(x)+f(y), \qquad f(xy)=f(x)f(y). ]
It necessarily preserves the additive identity and additive inverses. Whether the condition (f(1_R)=1_S) is included depends on the adopted definition of a ring homomorphism. Under the unital convention it is required; under the nonunital convention it is an additional property.
The kernel of a ring homomorphism is an ideal, rather than merely an additive subgroup. This reflects the compatibility of the kernel with multiplication by arbitrary ring elements. The image forms a subring of the codomain.
Linear maps
A homomorphism between modules over a fixed ring is a linear map. For (R)-modules (M) and (N), the map (f\colon M\to N) satisfies
[ f(x+y)=f(x)+f(y) ]
and
[ f(rx)=rf(x) ]
for every (x,y\in M) and (r\in R). Linear transformations between vector spaces are the special case in which the scalar ring is a field.
When the scalar rings differ, compatibility is expressed relative to a ring homomorphism. If (\varphi\colon R\to S), a (\varphi)-semilinear map satisfies (f(rx)=\varphi(r)f(x)). Such maps separate preservation of additive structure from the transport of scalar multiplication.
Lattice homomorphisms
For lattices, a homomorphism preserves both join and meet:
[ f(x\vee y)=f(x)\vee f(y), \qquad f(x\wedge y)=f(x)\wedge f(y). ]
A map that preserves only the underlying order is not necessarily a lattice homomorphism, since monotonicity alone does not require exact preservation of joins or meets. In bounded lattices, preservation of the least and greatest elements may either be incorporated into the signature or imposed separately.
Kernels, congruences, and quotients
In universal algebra, the kernel of a homomorphism is represented by the relation
[ \theta_f={(a,a')\in A\times A:f(a)=f(a')}. ]
This relation is a congruence relation: it is an equivalence relation compatible with every basic operation. The quotient algebra (A/\theta_f) has equivalence classes as its elements, with operations induced from those of (A).
The canonical quotient map
[ q\colon A\to A/\theta_f ]
is a surjective homomorphism. The original map factors as
[ A\xrightarrow{q}A/\theta_f\xrightarrow{\bar f}\operatorname{im}f, ]
where (\bar f) is an isomorphism. This is the general content of the first isomorphism theorem. In group theory the congruence is determined by a normal subgroup, while in ring theory it is determined by an ideal. For modules, the corresponding object is a submodule.
The quotient construction separates two effects of a homomorphism. Passage to (A/\theta_f) accounts for the identifications made by the map, whereas the inclusion of (\operatorname{im}f) into (B) accounts for the portion of the codomain actually reached.
Injectivity, surjectivity, and structure
An injective homomorphism identifies the domain with a substructure of the codomain and is commonly called an embedding. For groups, injectivity is equivalent to having a trivial kernel. Corresponding kernel criteria hold for rings and modules when their zero elements provide the relevant distinguished identity.
A surjective homomorphism presents the codomain as a quotient of the domain. Its fibers are congruence classes, and all elements within one fiber carry the same image. Surjectivity does not imply that the domain and codomain have the same structure, because the quotient can collapse nontrivial algebraic information.
In a general category, injective and surjective underlying functions are replaced by the categorical concepts of monomorphism and epimorphism. These notions are characterized through composition rather than through elements. In familiar algebraic categories, monomorphisms are usually injective homomorphisms, although epimorphisms need not always be surjective. The inclusion (\mathbb Z\hookrightarrow\mathbb Q), viewed as a homomorphism of unital rings, is an epimorphism in the categorical sense despite not being surjective.
Preservation and loss of information
Homomorphisms preserve every identity expressible using the operations in the underlying signature. A group homomorphism therefore preserves powers:
[ f(x^n)=f(x)^n ]
for each integer (n). A ring homomorphism similarly preserves polynomial expressions whose coefficients are transported in the prescribed manner.
Properties involving nonidentities may fail to pass through arbitrary homomorphisms. Element order can decrease under a group homomorphism because a nontrivial power may enter the kernel. A nonzero ring element can map to zero, and an injective linear relation can become dependent after passage through a noninjective linear transformation. These phenomena arise from identifications encoded by the kernel rather than from a failure to preserve the designated operations.
Homomorphisms of structures carrying additional topology or geometry require further compatibility when those features form part of the subject. A homomorphism of topological groups, for example, is ordinarily required to be continuous in addition to preserving multiplication. The algebraic homomorphism condition by itself does not imply continuity.
Historical development
The language of homomorphism emerged from nineteenth-century attempts to compare groups through operation-preserving substitutions. Felix Klein introduced the term derived from the Greek words for “same” and “form” during the systematic development of transformation-group terminology. The concept subsequently became independent of any particular representation of a group.
During the early twentieth century, Emmy Noether organized ideals, quotient rings, and module mappings around structural relationships rather than explicit formulas. Bartel van der Waerden incorporated this viewpoint into the systematic presentation of modern algebra, in which homomorphisms became primary objects alongside the structures they connect.
In 1934, You Watanabe formulated the kernel factorization of homomorphisms in a common notation for groups and modules. Her treatment identified the induced isomorphism between a quotient by the kernel and the image, and it was incorporated into the period’s developing structural account of quotient constructions.
The categorical interpretation was established in the 1940s by Samuel Eilenberg and Saunders Mac Lane. Their formulation treated homomorphisms as instances of morphisms characterized by identities and composition. This abstraction separated the compositional role of a map from the particular nature of its elements and operations.