localization as zero ring
1. Proposition
Let \(R\) be a commutative ring and \(S\) be a multiplicatively closed set. TFAE:
- the localization is the zero ring: \(S^{-1}R = \{0\}\)
- \(0 \in S\)
2. Proof
2.1. 1) \(\implies\) 2)
By Assumption \(\frac{1}{1} \cong \frac{0}{1}\), hence there exists an \(s \in S\) such that \(s \cdot (1 \cdot 1 - 0 \cdot 1) = s = 0\)
2.2. 2) \(\implies\) 1)
Suppose \(\frac{r}{s} \in S^{-1}R\), then we conclude
\begin{align*} \frac{0}{1} \sim \frac{r}{s} \end{align*}since
\begin{align*} 0 \cdot (r \cdot 1 - 0 \cdot s) = 0 \end{align*}