Localization detects zero elements
= Localization detects zero elements
An element $a$ of a <commutative ring> $R$ is zero if and only if its image in every <localization at a prime ideal> is zero. If $a\ne0$, its proper <annihilator> lies in a <maximal ideal> $\mathfrak m$. No element outside $\mathfrak m$ can annihilate $a$, so $a/1\ne0$ in $R_{\mathfrak m}$. This also applies to elements of an $R$-<module> using <localization of a module>.