# Algebra homomorphism

__: Ring homomorphism preserving scalar multiplication__

**Short description**It has been suggested that this article be merged into associative algebra. (Discuss) Proposed since March 2023. |

In mathematics, an **algebra homomorphism** is a homomorphism between two algebras. More precisely, if *A* and *B* are algebras over a field (or a ring) *K*, it is a function [math]\displaystyle{ F\colon A\to B }[/math] such that, for all *k* in *K* and *x*, *y* in *A*, one has^{[1]}^{[2]}

- [math]\displaystyle{ F(kx) = kF(x) }[/math]
- [math]\displaystyle{ F(x + y) = F(x) + F(y) }[/math]
- [math]\displaystyle{ F(xy) = F(x) F(y) }[/math]

The first two conditions say that *F* is a *K*-linear map, and the last condition says that *F* preserves the algebra multiplication. So, if the algebras are associative, F is a rng homomorphism, and, if the algebras are rings and F preserves the identity, it is a ring homomorphism.

If *F* admits an inverse homomorphism, or equivalently if it is bijective, *F* is said to be an isomorphism between *A* and *B*.

## Unital algebra homomorphisms

If *A* and *B* are two unital algebras, then an algebra homomorphism [math]\displaystyle{ F:A\rightarrow B }[/math] 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) ring homomorphism.

## Examples

- Every ring is a [math]\displaystyle{ \mathbb{Z} }[/math]-algebra since there always exists a unique homomorphism [math]\displaystyle{ \mathbb{Z} \to R }[/math]. See Associative algebra for the explanation.
- Any homomorphism of commutative rings [math]\displaystyle{ R \to S }[/math] gives [math]\displaystyle{ S }[/math] the structure of a commutative R-algebra. Conversely, if S is a commutative R-algebra, the map [math]\displaystyle{ r\mapsto r\cdot 1_S }[/math] is a homomorphism of commutative rings. It is straightforward to deduce that the overcategory of the commutative rings over R is the same as the category of commutative [math]\displaystyle{ R }[/math]-algebras.
- If
*A*is a subalgebra of*B*, then for every invertible*b*in*B*the function that takes every*a*in*A*to*b*^{−1}*a**b*is an algebra homomorphism (in case [math]\displaystyle{ A=B }[/math], this is called an inner automorphism of*B*). If*A*is also simple 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.

## See also

## References

- ↑ Dummit, David S.; Foote, Richard M. (2004).
*Abstract Algebra*(3rd ed.).*John Wiley & Sons*. ISBN 0-471-43334-9. - ↑ Lang, Serge (2002).
*Algebra*. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

Original source: https://en.wikipedia.org/wiki/Algebra homomorphism.
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