Multiplicative set

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

$1\in S$ , and
$s,t\in S$ implies $st\in S$ .

The definition permits $0\in S$ . In that case the localization $S^{-1}R$ 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

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.)

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 $R_{\mathfrak p}$ .

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 $p$ .

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Basic properties