Past exam of the algebra course of the University of Oxford / 2008 / A / 1 / i / Solution

An algebra homomorphism is a $K$-linear map preserving multiplication, and preserving the unit under the unital convention. An algebra isomorphism is a bijective algebra homomorphism; its inverse then preserves the same operations.

A commutative domain algebra may map into a noncommutative target: take $A=K$, $B=K\times M_2(K)$, and $\phi (\lambda )=(\lambda ,\lambda I_2)$. This is even an injective unital homomorphism. The matrix factor makes the target noncommutative over every field, since $E_{12}{E}_{21}\ne E_{21}{E}_{12}$.