---- > [!definition] Definition. ([[multiplicative subset of a ring]]) > Let $R$ be a [[commutative ring|commutative]] [[ring]]. A **multiplicative set** $S \subset R$ is a subset satisfying $1 \in S$ and $a,b\in S \implies ab \in S$. > If $U \subset R$ is any subset, its **multiplicative closure** is $S=\left\{ \prod_{i=1}^{n} u_{1} \cdots u_{n} : u_{i} \in U, n \geq 0 \right\}.$ ^definition > [!example] The three most important examples. > 1. If $R$ is an [[integral domain]], then $R-\{ 0 \}$ is a multiplicative set. > 2. $x \in R$ $\implies$ $S=\{ 1,x,x^{2},x^{3}, x^{4}, \dots \}$ is a multiplicative set, often denoted $R_{x}$. > 3. If $\mathfrak{p}$ is a [[prime ideal]] of $R$, then $R-\mathfrak{p}$ is a multiplicative set, often denoted $R_{\mathfrak{p}}$. ^example > [!basicexample] > > For $x \in R$, $S_{\{ x \}}=\{ x^{n}(1-rx): n \geq 0, r \in R \}$ is a multiplicative subset. Indeed, $1 \in S_{\{ x \}}$ (put $n=r=0$), and $\begin{align} > x^{n}(1-rx) \cdot x^{m}(1-sx) &= x^{n+m} (1- sx - rx + rs x^{2}) \\ > &= x^{n+m} (1-(r+s)x + rs x ^{2}) \\ > &= x^{n+m} \big( 1 - \big( (r+s) - rs x \big) x \big) \in S. > \end{align}$ > This is relevant in [[Krull dimension|dimension theory]]. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```