Algebra homomorphism

A homomorphism between two associative algebras, A and B, over a field (or commutative ring) K, is a function $F\colon A\to B$ such that for all k in K and x, y in A,^[1]^[2]

$F(kx)=kF(x)$
$F(x+y)=F(x)+F(y)$
$F(xy)=F(x)F(y)$

The first two conditions say that F is a K-module homomorphism between K-modules.

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

A common abbreviation for "homomorphism between algebras" is "algebra homomorphism" or "algebra map".

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.

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.

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[1] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.

[2] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

Algebra homomorphism

Unital algebra homomorphisms

Examples

See also

References