Minimal prime ideal: Difference between revisions
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==Definition== | ==Definition== | ||
An [[ideal]] in a [[commutative unital ring]] is termed a '''minimal prime ideal''' if it is a [[prime ideal]], and there is no prime ideal strictly contained inside it. | 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. | Note that for an [[integral domain]], the zero ideal is the unique minimal prime ideal. | ||
Revision as of 17:27, 3 March 2008
This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings
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.