I am confused about the polynomials and polynomial functions. I know when we say $R[x]$, we are refering to polynomials, so what exactly is the difference? Is it basically that one is a function and the other one isn't?
There's this section of Artin's book that discusses a map from polynomials to its associated polynomial function. The first line says that the existence of the homomorphism follows from the substitution principle. What does this mean? How does it follow?
Best Answer
It may help to think of the polynomial ring $R[X]$ differently, at least for a short moment, as the group $\bigoplus_{n\in\Bbb{N}}R$ with multiplication defined by $$(c_k)_{k\in\Bbb{N}}\cdot(d_k)_{k\in\Bbb{N}}=\left(\sum_{j=0}^kc_jd_{k-j}\right)_{k\in\Bbb{N}}.$$ Here of course the sequences $(c_k)_{k\in\Bbb{N}}$ and $(d_k)_{k\in\Bbb{N}}$ correspond to the polynomials $$\sum_{k\in\Bbb{N}}c_kX^k \qquad\text{ and }\qquad \sum_{k\in\Bbb{N}}d_kX^k,$$ where the sums are finite by definition of the direct product. The newly defined product of the sequences above then indeed corresponds to the product of these polynomials, so this ring is isomorphic to $R[X]$. In this ring the powers of the indeterminate $X$ correspond to the standard basis elements of the direct sum. They are in no way functions from $R$ to $R$.
Now every such sequence does define a function $R\ \longrightarrow\ R$ by substitution, i.e. by plugging in the elements of $R$. In this way the sequence $(c_k)_{k\in\Bbb{N}}\in R[X]$ defines the function $$R\ \longrightarrow\ R:\ r\ \longmapsto\ \sum_{k\in\Bbb{N}}c_kr^k.$$
Of course the same ideas work for the polynomial ring in $n$ indeterminates, by repeating this construction $n$ times.