definition of prime ideal by Artin
Lemma.β Let $R$ be a commutative ring and $S$ a multiplicative semigroup consisting of a subset of $R$.β If there exist http://planetmath.org/node/371ideals of $R$ which are disjoint with $S$, then the set $\mathrm{\pi \x9d\x94\x96}$ of all such ideals has a maximal element with respect to the set inclusion.
Proof.β Let $C$ be an arbitrary chain in $\mathrm{\pi \x9d\x94\x96}$.β Then the union
$$\mathrm{\pi \x9d\x94\x9f}:=\underset{\mathrm{\pi \x9d\x94\x9e}\beta \x88\x88C}{\beta \x8b\x83}\mathrm{\pi \x9d\x94\x9e},$$ |
which belongs to $\mathrm{\pi \x9d\x94\x96}$, may be taken for the upper bound of $C$, since it clearly is an ideal of $R$ and disjoint with $S$.β Because $\mathrm{\pi \x9d\x94\x96}$ thus is inductively ordered with respect to β$\beta \x8a\x86$β, our assertion follows from Zornβs lemma.
Definition.β The maximal elements in the Lemma are prime ideals^{} of the commutative ring.
The ring $R$ itself is always a prime ideal ($S=\mathrm{\beta \x88\x85}$).β If $R$ has no zero divisors^{}, the zero ideal^{} $(0)$ is a prime ideal ($S=R\beta \x88\x96\{0\}$).
If the ring $R$ has a non-zero unity element 1, the prime ideals corresponding the semigroup β$S=\{1\}$β are the maximal ideals^{} of $R$.
References
- 1 Emil Artin: Theory of Algebraic Numbers^{}.β Lecture notes.β 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 |