symmetric monoidal (∞,1)-category of spectra
Let $R$ be a commutative ring.
The set of polynomials in one variable with coefficients in $R$ is the set $R[\mathbb{N}]$ of all formal linear combinations on elements $n \in \mathbb{N}$, thought of as powers $x^n$ of the variable $x$
where $n$ is an arbitrary natural number and $a_0, \dots, a_n \in R$, modulo the equivalence relation generated by equations of the form
(so that we ignore coefficients of zero).
This set is equipped with the structure of a ring itself, in fact a commutative algebra over $R$, denoted $R[z]$ and called the polynomial ring or ring of polynomials given by the unique bilinear map
which on monomials is given by
In addition to the ring structure, there is a further operation $R[z] \times R[z] \to R[z]$ which may be described as “substitution”; see Remark 2 below for a general description (which applies in fact to any Lawvere theory).
Moreover, there is a noncommutative analogue of polynomial ring on a set $X$, efficiently described as the free $R$-module generated by the (underlying set of the) free monoid on $X$. This carries also a ring structure, with ring multiplication induced from the monoid multiplication. A far-reaching generalization of this construction is given at distributive law.
Finally: polynomial algebras may be regarded as graded algebras (graded over $\mathbb{N}$). Specifically: let us regard $R[X]$ as the free $R$-module generated by (the underlying set of) the free commutative monoid $F(X)$. The monoid homomorphism $F(!): F(X) \to F(1) \cong \mathbb{N}$ induced by the unique function $!: X \to 1$ gives an $\mathbb{N}$-fibering of $F(X)$ over $\mathbb{N}$, with typical fiber $F(X)_n$ whose elements are called monomials of degree $n$. Then the homogeneous component of degree $n$ in $R[X]$ is the $R$-submodule generated by the subset $F(X)_n \subset F(X)$. The elements of this component are called homogeneous polynomials of degree $n$.
By the definition of free objects one needs to check that ring homomorphisms
to another ring K are in natural bijection with functions of sets
from the singleton to the set underlying $K$. Take $\bar f \coloneqq f(z)$. Using $R$-linearity, this is directly seen to yield the desired bijection.
Similarly, the set of polynomials in any give set of variables with coefficients in $R$ is the free commutative $R$-algebra on that set of generators; see symmetric power and symmetric algebra.
As usual in the study of universal algebra via Lawvere theories, there is an operad whose $n^{th}$ component $C_n$ is the free algebra $R[x_1, \ldots, x_n]$, and whose operadic multiplication is given by maps
($n = n_1 + \ldots + n_k$) that take a tuple of elements $(p; q_1, \ldots, q_k)$ to $p(q_1(x), \ldots, q_k(x))$. Formally, it takes this tuple to the value of $p$ under the unique algebra map $R[x_1, \ldots, x_k] \to R[x_1, \ldots, x_n]$ that extends the mapping $x_j \mapsto i_j(q_j)$. Here the $i_j: R[x_1, \ldots, x_{n_j}] \to R[x_1, \ldots, x_n]$ are appropriate coproduct inclusions (in the category of commutative rings), where $i_j(x_l) = x_{n_1 + \ldots + n_{j-1} + l}$. A particularly important case of substitution is the case $k=1$ and $n_1 = 1$, where the map $R[x] \times R[x] \to R[x]$ is ordinary substitution $(p, q) \mapsto p(q(x))$. This is a special case of the more general notion of Tall-Wraith monoid.
In case $R$ is an integral domain, the field of fractions of $R[z]$ is the field $R(z)$ of rational functions.