# Algebra homomorphism

Short description: Ring homomorphism preserving scalar multiplication

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 $\displaystyle{ F\colon A\to B }$ such that, for all k in K and x, y in A, one has[1][2]

• $\displaystyle{ F(kx) = kF(x) }$
• $\displaystyle{ F(x + y) = F(x) + F(y) }$
• $\displaystyle{ F(xy) = F(x) F(y) }$

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 $\displaystyle{ F:A\rightarrow B }$ 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) ring homomorphism.

## Examples

• Every ring is a $\displaystyle{ \mathbb{Z} }$-algebra since there always exists a unique homomorphism $\displaystyle{ \mathbb{Z} \to R }$. See Associative algebra for the explanation.
• Any homomorphism of commutative rings $\displaystyle{ R \to S }$ gives $\displaystyle{ S }$ the structure of a commutative R-algebra. Conversely, if S is a commutative R-algebra, the map $\displaystyle{ r\mapsto r\cdot 1_S }$ is a homomorphism of commutative rings. It is straightforward to deduce that the overcategory of the commutative rings over R is the same as the category of commutative $\displaystyle{ R }$-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 $\displaystyle{ A=B }$, 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.