TITLE: Importance and Intuition of Polynomial Rings
QUESTION [18 upvotes]: I know that a polynomial ring $R[x]$ is the ring with elements consisting of polynomials with coefficients in $R$. However, this definition leaves me confused when I try to really understand the concept of polynomial rings, rather than just accept that definition for what it is. Where did the polynomials come from? Why are polynomial rings important? Is there a way to “construct” the polynomials in the context of Abstract Algebra, so that we may see why studying rings made up of polynomials is important? Are polynomial rings actually just sets of polynomials, or are the polynomials just a concrete way to represent some more abstract idea? What does the variable $x$ mean in polynomial rings? I apologize for all the questions, I’m just having trouble seeing the importance or intuition behind them.

REPLY [7 votes]: I will elaborate my answer on how to see the polynomial ring without the indeterminate $x,$ and on the universal property of polynomial rings. I will show you two alternative ways to understand the definition of polynomial rings. Those interpretations will allow you to see this construction without wondering about the role of indeterminate(s). Let $R$ be a commutative ring with unity $1_R.$

Monoid Ring on the Natural Numbers $R[\mathbb{N}]$:
This is the monoidal version of the group ring construction over the monoid of natural numbers. Here objects are formal finite linear combinations of natural numbers with coefficients in $R.$ For example $$r_1n_1+r_2n_2+\cdots+r_jn_j,\qquad r_1, r_2, \cdots, r_j\in R,\quad n_1, n_2,\cdots, n_j\in\mathbb{N}.$$ It is not difficult to see that this is another way to look at the polynomial ring by interpreting above element as the polynomial $$r_1x^{n_1}+r_2x^{n_2}+\cdots+r_jx^{n_j}\in R[x].$$ Furthermore, if you use $\mathbb{N}^n$ (ordered $n$-tuples of natural numbers) in this construction, you would get the  polynomial ring $R[x_1, x_2,\cdots, x_n]$ with $n$ variables.

Ring of $R$-sequences of finite support $R^{(\mathbb{N})}$:
Another way to look at the linear combination $r_1n_1+r_2n_2+\cdots+r_jn_j$ is as a function $$f:\mathbb{N}\to R$$ whose values are $f(n_i)=r_i$ for all $i=1, 2, \cdots, j$ and zero everywhere else. Since the domain of this function is non-negative integers, this is nothing but a sequence. Also, the sequence is zero for all but finitely many terms, therefore we say it is finitely supported. On the other hand if you are given a such sequence we can easily create a unique polynomial (and a linear combinaion). For example: Say $R=\mathbb{Z}/3\mathbb{Z},$ then the finitely supported sequence $(1, 1, 1, 0, 0, 0, \cdots)$ represents the polynomial $1+x+x^2=(2+x)^2$ and vise versa. Note that even this factorization carries all the information about the associated sequence.


Above two equivalent (ring isomorphic) constructions shows us that powers of indeterminate $x$ are nothing but placeholders. We can identify $x$ with the linear combination $1_R1$ or with the sequence $(0, 1_R, 0, 0, \cdots).$ Also, if we accept any of them as the polynomial ring, say the first construction, then we have an injective ring homomorphism (embedding) $$\iota : R \to R[\mathbb{N}]$$ given by $r\mapsto r0$ (or in the other construction $r\mapsto (r, 0, 0, 0, \cdots)\in R^{(\mathbb{N})}$). This map is universal in the sence that any other such map factor through $\iota$ while sending $x$ to a priscribe value. This is the universal property of polynomial ring (of a single variable) over $R$:

For every commutative unital ring $S$ that has an injective ring homomorphism $R\xrightarrow{\phi}S$ and for any chosen $a\in S,$ there is a unique extension (evaluation homomorphism) $R[\mathbb{N}]\xrightarrow{\hat\phi_a}S$ such that $\hat\phi_a\circ\iota=\phi$ and $\hat\phi_a(x)=a.$
$$\require{AMScd}
\require{cancel}
\def\diaguparrow#1{\smash{\raise.6em\rlap{\ \ \scriptstyle #1}
   \lower.6em{\cancelto{}{\Space{2em}{1.7em}{0px}}}}}
\begin{CD}
&& R[\mathbb{N}]\\
& \diaguparrow{\iota} @VV\hat\phi_a V \\
R @>>\phi> S
\end{CD}$$

One can easily formulate similar universal property for polynomial ring with any number of variables.