Multiplicative set

Let $R$ be a commutative ring . A subset $S\subseteq R$ is a multiplicative set if

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

Often one also assumes $0\notin S$ when the goal is to form the localization of a ring $S^{-1}R$; if $0\in S$, then $0$ becomes invertible in $S^{-1}R$, forcing $1=0$ and hence $S^{-1}R$ is the zero ring.

A key source of multiplicative sets is complements of primes: if $\mathfrak p\subset R$ is prime, then $R\setminus \mathfrak p$ is multiplicative, and this choice produces the localization at a prime .

Examples

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

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

   is multiplicative. (If $f$ is nilpotent, then $0\in S$ and the corresponding localization collapses to the zero ring.)

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

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

   is multiplicative (primality ensures $st\notin\mathfrak p$ whenever $s,t\notin\mathfrak p$). Localizing at this $S$ gives $R_{\mathfrak p}$.

3. Inverting a prime number in $\mathbb Z$. In $R=\mathbb Z$, the subset $S=\{1,p,p^2,\dots\}$ (for a prime $p$) is multiplicative. The localization $S^{-1}\mathbb Z$ is the subring of $\mathbb Q$ consisting of fractions whose denominator is a power of $p$.