definition of prime ideal by Artin

Lemma.  Let R be a commutative ring and S a multiplicative semigroup consisting of a subset of R.  If there exist of R which are disjoint with S, then the set 𝔖 of all such ideals has a maximal element with respect to the set inclusion.

Proof.  Let C be an arbitrary chain in 𝔖.  Then the union


which belongs to 𝔖, may be taken for the upper bound of C, since it clearly is an ideal of R and disjoint with S.  Because 𝔖 thus is inductively ordered with respect to β€œβŠ†β€, our assertion follows from Zorn’s lemma.

Definition.  The maximal elements in the Lemma are prime idealsMathworldPlanetmathPlanetmathPlanetmath of the commutative ring.

The ring R itself is always a prime ideal (S=βˆ…).  If R has no zero divisorsMathworldPlanetmath, the zero idealMathworldPlanetmathPlanetmath (0) is a prime ideal (S=Rβˆ–{0}).

If the ring R has a non-zero unity element 1, the prime ideals corresponding the semigroup  S={1}  are the maximal idealsMathworldPlanetmath of R.


  • 1 Emil Artin: Theory of Algebraic NumbersMathworldPlanetmath.  Lecture notes.  Mathematisches Institut, GΓΆttingen (1959).
Title definition of prime ideal by Artin
Canonical name DefinitionOfPrimeIdealByArtin
Date of creation 2013-03-22 18:44:31
Last modified on 2013-03-22 18:44:31
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Definition
Classification msc 13C99
Classification msc 06A06
Related topic EveryRingHasAMaximalIdeal
Defines prime ideal