Let $R$ be a finitely generated ring and $M$ a finite minimal generating set for $R$.

I'm looking for a theorem that states that a function $\varphi:M \rightarrow S$ extends to a unique ring homomorphism $\varphi: R \rightarrow S$. Does such a theorem exist? Or possibly a similar one? I am aware of the universal mapping property of groups and am looking for one analgous for rings.