Consider $R = k[x, y]$ and $f = x$. The localization $R_f = k[x, y]_x$ consists of rational functions $p(x,y)/x^n$ where $p$ is a polynomial.

Geometrically, $R_f$ corresponds to the complement of the variety $V(x) = \{(0, y) : y \in k\}$ in the affine plane. We are studying functions away from the $y$-axis.

More generally, $k[x_1, \ldots, x_n]_f$ corresponds to the open set $D(f) = \text{Spec}(R) \setminus V(f)$.

For a prime $p$, the localization $\mathbb{Z}_{(p)}$ consists of fractions $a/b$ where $p \nmid b$: $\mathbb{Z}_{(p)} = \left\{\frac{a}{b} : a, b \in \mathbb{Z}, \gcd(b,p) = 1\right\}$

This is a local ring with maximal ideal $(p) = p\mathbb{Z}_{(p)}$. The quotient $\mathbb{Z}_{(p)}/(p) \cong \mathbb{F}_p$ is the residue field.

The localization "focuses attention" on divisibility properties related to $p$, ignoring other primes.

For an affine variety $V \subseteq \mathbb{A}^n$, the function field $k(V)$ is the field of fractions of the coordinate ring $k[V] = k[x_1, \ldots, x_n]/I(V)$: $k(V) = \text{Frac}(k[V])$

Elements are rational functions $f/g$ where $f, g \in k[V]$ and $g$ is not identically zero on $V$. This captures the idea of "rational functions" on the variety.

If $I = Q_1 \cap \cdots \cap Q_n$ is a primary decomposition with associated primes $\mathfrak{p}_1, \ldots, \mathfrak{p}_n$, then: $S^{-1}I = \bigcap_{\mathfrak{p}_i \cap S = \emptyset} S^{-1}Q_i$

Localizing "removes" primary components whose associated primes meet $S$. For instance, localizing at $\mathfrak{p}$ isolates the $\mathfrak{p}$-primary component.

For $R = k[x]$ and $S = \{1, x, x^2, \ldots\}$, the localization: $R_S = k[x]_x = k[x, x^{-1}]$ is the ring of Laurent polynomials, consisting of finite sums $\sum_{i=m}^n a_i x^i$ where $m$ can be negative.

This appears in studying functions on the punctured line $\mathbb{A}^1 \setminus \{0\}$, or in algebraic contexts like quantum groups.