Universal Property of the Polynomial Ring

theoremring-theory

Intuition

Formal Statement

For $f:R\to S$ a morphism. If we fix $\bar{s}\in S$, then there exists a unique $F:R[x]\to S$ characterized by,

• $F{\mid }_R=f$ ($F$ is equivalent to $f$ when restricted to $R$).
• $F(x)=\bar{s}$.

In fact, $F(a_0+a_{1}x+\dots +a_n{x}^n)=f(a_0)+f(a_1)\bar{s}+f(a_2){\bar{s}}^2+\dots +f(a_n){\bar{s}}^n$. In particular, if $\underset{f={\mathrm{id}}_f}{R=S}$ and $\bar{r}\in R$, we can evaluate $P(x)$ at $x=\bar{r}$ by $P(\bar{r})=a_0+a_1\bar{r}+\dots +a_n{\bar{r}}^n$.

Corollaries

• $P(x)\in R[x]⇝{\phi }_p:R\to R$ given by ${\phi }_p(r)=P(r)$ polynomial functions. Caution: it can happen that different polynomials have same polynomial function.