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
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.
The field of fractions of $R[z]$ is the field $R(z)$ of rational functions.