In mathematics, an algebra homomorphism is a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

between two associative algebras. More precisely, if and are

algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...

over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

(or commutative ring) , it is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

F\colon A\to B

such that for all in and in , *

F(kx) = kF(x)

F(x + y) = F(x) + F(y)

F(xy) = F(x) F(y)

The first two conditions say that is a ''K''-

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

(or ''K''-module homomorphism if ''K'' is a commutative ring), and the last condition says that is a (non-unital)

ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...

. If admits an inverse homomorphism, or equivalently if it is

bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

, is said to be an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

between and .

Unital algebra homomorphisms

If ''A'' and ''B'' are two unital algebras, then an algebra homomorphism

F:A\rightarrow B

is said to be ''unital'' if it maps the unity of ''A'' to the unity of ''B''. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded. A unital algebra homomorphism is a (unital)

Examples

* Every ring is a

\mathbb

-algebra since there always exists a unique homomorphism

\mathbb \to R

. See Associative algebra#Examples for the explanation. * Any homomorphism of commutative rings

R \to S

gives

S

the structure of a commutative -algebra. Conversely, if is a commutative -algebra, the map

r\mapsto r\cdot 1_S

is a homomorphism of commutative rings. It is straightforward to deduce that the overcategory of the commutative rings over is the same as the category of commutative

R

-algebras. * If ''A'' is a

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...

of ''B'', then for every

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

''b'' in ''B'' the function that takes every ''a'' in ''A'' to ''b''⁻¹ ''a'' ''b'' is an algebra homomorphism (in case

A=B

, this is called an inner automorphism of ''B''). If ''A'' is also

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

and ''B'' is a central simple algebra, then every homomorphism from ''A'' to ''B'' is given in this way by some ''b'' in ''B''; this is the

Skolem–Noether theorem In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in ...

References

{{DEFAULTSORT:Algebra Homomorphism Algebras Ring theory Morphisms

Unital algebra homomorphisms

Examples

See also

References