In the context of ring theory, a **minimal prime ideal** is a prime ideal \( P \) in a commutative ring \( R \) such that there are no other prime ideals contained within \( P \) except for \( P \) itself. In other words, \( P \) is a minimal element in the set of prime ideals of the ring with respect to inclusion.
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A minimal prime ideal is a prime ideal containing no strictly smaller prime ideal. If is a reduced ring and is minimal, the localization at a prime ideal 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.