Multiplicative set

Let $R$ be a commutative ring (with $1$). A multiplicative set (or multiplicative subset) is a subset $S \subseteq R$ such that:

1. $1 \in S$,
2. If $s,t \in S$ then $st \in S$.

Often (especially in commutative algebra) one also assumes $0 \notin S$; if $0 \in S$ then the corresponding localization $S^{-1}R$ collapses to the zero ring.

A key source of multiplicative sets is complements of prime ideals , which define localization at a prime .

Examples

1. Powers of one element. For $f \in R$, the set

   $$S=\{1,f,f^2,f^3,\dots\}$$

   is multiplicative. Localizing at $S$ “inverts $f$”.

2. Complement of a prime ideal. If $\mathfrak p \subset R$ is a prime ideal , then

   $$S = R \setminus \mathfrak p$$

   is multiplicative. This is the standard choice for building $R_{\mathfrak p}$.

3. Nonzero integers. In $R=\mathbb Z$, the set $S=\mathbb Z\setminus\{0\}$ is multiplicative; localizing gives the field of fractions $\mathbb Q$.