# Algebra homomorphism

An algebra homomorphism is an homomorphism between two associative algebras. More precisely, if A and B, are algebras over a field (or commutative ring) K, it is a function $F\colon A\to B$ such that for all k in K and x, y in A,

• $F(kx)=kF(x)$ • $F(x+y)=F(x)+F(y)$ • $F(xy)=F(x)F(y)$ The first two conditions say that F is a module 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 $F:A\rightarrow B$ 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 in the meaning of "unital algebra homomorphism", so non-unital algebra homomorphisms are excluded.

A unital algebra homomorphism is a ring homomorphism.

## Examples

• Every ring is a $\mathbb {Z}$ -algebra since there always exists a unique homomorphism $\mathbb {Z} \to R$ . See Associative algebra#Examples for the explanation.
• Any homomorphism of commutative rings $R\to S$ gives $S$ the structure of an $R$ -algebra. It is easy to use this to show that the overcategory $R/{\textbf {CRings}}$ is the same as the category of $R$ -algebras.
• Consider the diagram of $\mathbb {C} [z]$ -algebras
${\begin{matrix}&&\mathbb {C} [z]&\\&\swarrow {\operatorname {ev} _{\alpha }}&&\searrow \\\mathbb {C} &&\rightarrow &\mathbb {C} [x,y,z]/(xy-z)\end{matrix}}$ where $\operatorname {ev} _{\alpha }(z)=\alpha$ . This is

• 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 $A=B$ , 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.