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

where ${\displaystyle \operatorname {ev} _{\alpha }(z)=\alpha }$ . 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 ${\displaystyle A=B}$ , 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.