Krull's Theorem

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.
* 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

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

References

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*{{cite journal , first=W. , last=Hodges , title=Krull implies Zorn , journal=Journal of the London Mathematical Society , volume=s2-19 , issue=2 , year=1979 , pages=285-287 , doi=10.1112/jlms/s2-19.2.285

Ideals (ring theory)