Minimal prime ideal

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This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki
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.

Relation with other properties

Weaker properties