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