Minimal prime ideal

Definition
An ideal in a commutative unital ring is termed a minimal prime ideal if it satisfies the following equivalent conditions:


 * It is a prime ideal, and there is no prime ideal strictly contained inside it
 * The corresponding closed subset in the spectrum of the ring is a maximal irreducible closed subset: in other words, it is an irreducible closed subset not contained in any bigger irreducible closed subset.

Note that for an integral domain, the zero ideal is the unique minimal prime ideal.

Weaker properties

 * Prime ideal