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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*.

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.

- 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*⋅ 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*-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.