Algebra homomorphism

Unital algebra homomorphisms

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 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

• 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. 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.