Algebra homomorphism

In mathematics, 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 K-linear map (or K-module homomorphism if K is a commutative ring), and the last condition says that F is a (non-unital) 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#Examples 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.

See also

• Morphism
• Universal algebra § Basic constructions
• Spectrum of a ring
• Augmentation (algebra)

References

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.