Let $R$ be a commutative ring with unity, I have seen that
$A$ is a finitely generated $R$-algebra if and only if it is isomorphic to a quotient ring of the form $R[X_1,\dots,X_n]/I$ by an ideal $I\subset R[X_1,\dots,X_n]$.
Here when they say isomorphic, are we talking about homomorphisms of rings? or does it have to be an $R$ algebra homomorphism? Thank you.