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.
On
Thanks to egreg for pointing out that such a theorem does not exist, however there is a theorem for polynomial rings that is similar to the one I was searching for. See lemma 21.3: http://math.mit.edu/~mckernan/Teaching/12-13/Spring/18.703/l_21.pdf
This result can be extended to polynomial rings in multiple variables by induction.
Such a universal property characterizes free objects (here, rings).
In the case of commutative rings, the property that a ring $R$ has a (finite) subset $M$ so that every map $M\to S$ (where $S$ is any commutative ring) uniquely extends to a ring homomorphism $R\to S$ is the same as saying that $R$ is a polynomial ring over $\mathbb{Z}$ in $|M|$ indeterminates.
For instance, if $R=\mathbb{Z}/4\mathbb{Z}$, any subset is a set of generators; on the other hand there is no ring homomorphism $\varphi\colon\mathbb{Z}/4\mathbb{Z}\to\mathbb{Z}/3\mathbb{Z}$.
If two elements in $M$ satisfy a polynomial relation with integer coefficients, then also their images under $\varphi$ should satisfy the same relation in order that $\varphi$ can extend to a ring homomorphism.