Minimal prime ideal
= Minimal prime ideal
A minimal <prime ideal> is a <prime ideal> containing no strictly smaller <prime ideal>. If $A$ is a <reduced ring> and $\mathfrak p$ is minimal, the <localization at a prime ideal> $A_{\mathfrak p}$ is a field: its only prime is its <maximal ideal>, and reducedness makes that <ideal> its zero <nilradical>. Every nonzero <commutative ring> has a minimal prime.