Multiplicative set

Remarks

Example: let $\mathfrak p$ be a prime ideal of $R$ and $S = R \setminus \mathfrak p$. Then $1 \in S$ because $\mathfrak p$ is a proper ideal. If $s, t \notin \mathfrak p$, then $st \notin \mathfrak p$ by the definition of a prime ideal, so $st \in S$.

Other examples are the powers $\set{1, f, f^2, \ldots}$ of an element $f \in R$, the units of $R$, and the set of elements of $R$ that are not zero divisors.

Non-example: in $\Z$, the set $S = R \setminus 6\Z$ is not multiplicative. It contains $2$ and $3$ but not $2 \cdot 3 = 6$. The ideal $6\Z$ is not prime.

Some authors also require $0 \notin S$. If $0 \in S$, the localization $S^{-1}R$ is the zero ring.