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.