algebra homomorphism

For algebras $A$ and $B$, a map $\phi :A\to B$ is an algebra homomorphism if it satisfies the following conditions:

1. $\phi$ is linear
2. $\phi$ respects multiplication in the algebra

$$\phi (ab)=\phi (a)\phi (b)$$

3. $\phi (1)=1_B$

If there exists an inverse morphism then $\phi$ is called an isomorphism. Note, for this category, it suffices to check that $\phi$ is a bijective homomorphism to prove it is an isomorphism.

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