In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function such that for all in and in , The first two conditions say that is a K-linear map (or K-module homomorphism if K is a commutative ring), and the last condition says that is a (non-unital) ring homomorphism. If admits an inverse homomorphism, or equivalently if it is bijective, is said to be an isomorphism between and . 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 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 -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 -algebra. Conversely, if is a commutative -algebra, the map is a homomorphism of commutative rings. It is straightforward to deduce that the overcategory of the commutative rings over is the same as the category of commutative -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 , 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 Spectrum of a ring Augmentation (algebra) References Algebras Ring theory Morphisms