Algebra homomorphism

In mathematics, an algebra homomorphism is a homomorphism between two algebras. More precisely, if A and B are algebras over a field (or a ring) K, it is a function F : A → B such that, for all k in K and x, y in A, one has[1][2]

The first two conditions say that F is a K-linear map, and the last condition says that F preserves the algebra multiplication. So, if the algebras are associative, F is a rng homomorphism, and, if the algebras are rings and F preserves the identity, it is a 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 F : A → 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 Z-algebra since there always exists a unique homomorphism Z → R. See Associative algebra § Examples for the explanation.
  • Any homomorphism of commutative rings R → S gives S the structure of a commutative R-algebra. Conversely, if S is a commutative R-algebra, the map r ↦ r ⋅ 1S 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 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 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

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.
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