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 to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded.
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 a commutative R-algebra. Conversely, if S is a commutative R-algebra, the map 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 -algebras.
If A is a subalgebra of B, then for every invertibleb in B the function that takes every a in A to b−1ab 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.
algebra, homomorphism, mathematics, algebra, homomorphism, homomorphism, between, associative, algebras, more, precisely, algebras, over, field, commutative, ring, function, displaystyle, colon, such, that, displaystyle, displaystyle, displaystyle, first, cond. In mathematics an algebra homomorphism is a homomorphism between two associative algebras More precisely if A and B are algebras over a field or commutative ring K it is a function F A B displaystyle F colon A to B such that for all k in K and x y in A 1 2 F k x k F x displaystyle F kx kF x F x y F x F y displaystyle F x y F x F y 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 Contents 1 Unital algebra homomorphisms 2 Examples 3 See also 4 ReferencesUnital algebra homomorphisms EditIf A and B are two unital algebras then an algebra homomorphism F A B 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 EditEvery ring is a Z displaystyle mathbb Z algebra since there always exists a unique homomorphism Z R displaystyle mathbb Z to R See Associative algebra Examples for the explanation Any homomorphism of commutative rings R S displaystyle R to S gives S displaystyle S the structure of a commutative R algebra Conversely if S is a commutative R algebra the map r r 1 S 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 R 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 A B 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 EditMorphism Universal algebra Basic constructions Spectrum of a ring Augmentation algebra References Edit Dummit David S Foote Richard M 2004 Abstract Algebra 3rd ed John Wiley amp Sons ISBN 0 471 43334 9 Lang Serge 2002 Algebra Graduate Texts in Mathematics Springer ISBN 0 387 95385 X Retrieved from https en wikipedia org w index php title Algebra homomorphism amp oldid 1014718316, wikipedia, wiki, book, books, library,
