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 such that for all k in K and x, y in A,[1][2]

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 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.


  • Every ring is a -algebra since there always exists a unique homomorphism . See Associative algebra#Examples for the explanation.
  • Any homomorphism of commutative rings gives the structure of an -algebra. It is easy to use this to show that the overcategory is the same as the category of -algebras.
  • Consider the diagram of -algebras

where . 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 , 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


  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.
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