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 http://planetmath.org/node/371ideals of $R$ which are disjoint with $S$ , then the set $\mathfrak{S}$ of all such ideals has a maximal element with respect to the set inclusion.

Proof. Let $C$ be an arbitrary chain in $\mathfrak{S}$ . Then the union

\mathfrak{b}\;:=\;\bigcup_{\mathfrak{a}\in C}\mathfrak{a},

which belongs to $\mathfrak{S}$ , may be taken for the upper bound of $C$ , since it clearly is an ideal of $R$ and disjoint with $S$ . Because $\mathfrak{S}$ thus is inductively ordered with respect to “ $\subseteq$ ”, our assertion follows from Zorn’s lemma.

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

The ring $R$ itself is always a prime ideal ( $S=\varnothing$ ). If $R$ has no zero divisors, the zero ideal $(0)$ is a prime ideal ( $S=R\!\smallsetminus\!\{0\}$ ).

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

References

1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (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