[[Algebra theory MOC]]
# Algebra homomorphism

Let $A$ and $B$ be $R$-[[R-algebra|algebras]]
An **algebra homomorphism** $f : A \to B$ is a $R$-[[linear map]] which preserves the bilinear product, #m/def/falg 
i.e. for $x,y \in A$ and $\lambda,\mu \in R$
1. $f(\lambda x + \mu y)= \lambda f(x) + \mu f(y)$ ^H1
2. $f(xy)=f(x)f(y)$ ^H2

a **unital algebra homomorphism** is one which additionally has

3. $f(1_{A}) = 1_{B}$

## Properties

- [[Kernel of an algebra homomorphism]]

## Special cases

- [[Lie algebra homomorphism]]

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#state/tidy  | #lang/en | #SemBr