Dominating decision rule

In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn's lemma, which in turn is equivalent to the axiom of choice.

Variants

• For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold.
• For pseudo-rings, the theorem holds for regular ideals.Template:Disambiguate
• A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows:

Let R be a ring, and let I be a proper ideal of R. Then there is a maximal ideal of R containing I.

This result implies the original theorem, by taking I to be the zero ideal (0). Conversely, applying the original theorem to R/I leads to this result.

To prove the stronger result directly, consider the set S of all proper ideals of R containing I. The set S is nonempty since I ∈ S. Furthermore, for any chain T of S, the union of the ideals in T is an ideal J, and a union of ideals not containing 1 does not contain 1, so J ∈ S. By Zorn's lemma, S has a maximal element M. This M is a maximal ideal containing I.

Krull's Hauptidealsatz

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Another theorem commonly referred to as Krull's theorem:

Let $R$ be a Noetherian ring and $a$ an element of $R$ which is neither a zero divisor nor a unit. Then every minimal prime ideal $P$ containing $a$ has height 1.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathematische Annalen 10 (1929), 729–744.