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\subsection{Definitions} | ||
\begin{definition} | ||
A \emph{multiplicative set} or \emph{multiplicatively closed set} \( S \subseteq R \) is a subset such that \( 1 \in S \) and if \( a, b \in S \), then \( ab \in S \). | ||
If \( U \subseteq R \) is any set, its \emph{multiplicative closure} \( S \) of \( U \) is the set | ||
\[ \qty{\prod_{i = 1}^n u_i \mid n \geq 0, u_i \in U} \] | ||
which is the smallest multiplicatively closed set containing \( U \). | ||
\end{definition} | ||
\begin{example} | ||
\begin{enumerate} | ||
\item If \( R \) is an integral domain, then \( S = R \setminus \qty{0} \) is multiplicative. | ||
\item More generally, if \( \mathfrak p \) is a prime ideal in \( R \), then \( S = R \setminus \mathfrak p \) is multiplicative. | ||
\item If \( x \in R \), then the set \( \qty{x^n \mid n \geq 0} \) is multiplicative. | ||
\end{enumerate} | ||
\end{example} | ||
\begin{remark} | ||
\( \mathbb Q \) is obtained from \( \mathbb Z \) by adding inverses for the elements of the multiplicative subset \( \mathbb Z \setminus \qty{0} \). | ||
We have a ring homomorphism \( \mathbb Z \hookrightarrow \mathbb Q \). | ||
We generalise this construction to arbitrary rings and multiplicative sets. | ||
In general, injectivity of the ring homomorphism in question may fail. | ||
\end{remark} | ||
\begin{definition} | ||
Let \( S \subseteq R \) be a multiplicative set, and let \( M \) be an \( R \)-module. | ||
Then the \emph{localisation} of \( M \) by \( S \) is the set \( S^{-1} M = \faktor{M \times S}{\sim} \) where \( (m_1, s_1) \sim (m_2, s_2) \) if and only if there exists \( u \in S \) such that \( u(s_2 m_1 - s_1 m_2) = 0 \). | ||
We write \( \frac{m}{s} \) for the equivalence class corresponding to \( (m, s) \). | ||
We make \( S^{-1} M \) into an \( R \)-module by defining | ||
\[ \frac{m_1}{s_1} + \frac{m_2}{s_2} = \frac{m_1 s_2 + m_2 s_1}{s_1 s_2};\quad r \cdot \frac{m}{s} = \frac{rm}{s} \] | ||
We can make \( S^{-1} R \) into a ring by defining | ||
\[ \frac{r_1}{s_1} \cdot \frac{r_2}{s_2} = \frac{r_1 r_2}{s_1 s_2} \] | ||
\end{definition} |
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