Finite morphism

Definition
Suppose $$R$$ and $$S$$ are commutative unital rings and $$f:R \to S$$ is a homomorphism of commutative unital rings. This makes $$S$$ naturally into a $$R$$-module. Then, $$f$$ is termed finite if $$S$$ is a finitely generated module over $$R$$.

When the morphism is injective, we say that $$S$$ is a finite extension of $$R$$.