prime ideals by Artin are prime ideals

Theorem.  Due to Artin, a prime idealMathworldPlanetmathPlanetmathPlanetmath of a commutative ring R is the maximal elementMathworldPlanetmath among the ideals not intersecting a multiplicative subset S of R.  This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( to the usual criterion

a⁒bβˆˆπ”­β€ƒβ‡’β€ƒaβˆˆπ”­βˆ¨bβˆˆπ”­ (1)

of prime ideal (see the entry prime ideal (

Proof.  1oΒ―.  Let 𝔭 be a prime ideal by Artin, corresponding the semigroup S, and let the ring product a⁒b belong to 𝔭.  Assume, contrary to the assertion, that  neither of a and b lies in 𝔭.  When  (𝔭,x)  generally means the least ideal containing 𝔭 and an element x, the antithesis implies that


whence by the maximality of 𝔭 we have

(𝔭,a)∩Sβ‰ βˆ…βˆ§(𝔭,b)∩Sβ‰ βˆ….

Therefore we can chose such elements  si=pi+ri⁒a+ni⁒a  of S (N.B. the multiples) that

piβˆˆπ”­,ri∈R,niβˆˆβ„€β€ƒ(i= 1, 2).

But then


This is however impossible, since the productPlanetmathPlanetmath s1⁒s2 belongs to the semigroup S and  π”­βˆ©S=βˆ….  Because the antithesis thus is wrong, we must have  aβˆˆπ”­β€‰ or  bβˆˆπ”­.

2oΒ―.  Let us then suppose that an ideal 𝔭 satisfies the condition (1) for all  a,b∈R.  It means that the set  S=Rβˆ–π”­β€‰ is a multiplicative semigroup.  Accordingly, the 𝔭 is the greatest ideal not intersecting the semigroup S, Q.E.D.

Remark.  It follows easily from the theorem, that if 𝔭 is a prime ideal of the commutative ring 𝔒 and 𝔬 is a subring of 𝔒, then π”­βˆ©π”¬ is a prime ideal of 𝔬.

Title prime ideals by Artin are prime ideals
Canonical name PrimeIdealsByArtinArePrimeIdeals
Date of creation 2013-03-22 18:44:55
Last modified on 2013-03-22 18:44:55
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 13C99
Classification msc 06A06
Related topic IdealGeneratedByASet
Related topic PrimeIdeal