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Krull's theorem

Part of ring theory in mathematics From Wikipedia, the free encyclopedia

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In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring[1] 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,[2] which in turn is equivalent to the axiom of choice.

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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.
  • An apparently 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.
The statement of the original theorem can be obtained 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 IS. 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 JS. By Zorn's lemma, S has a maximal element M. This M is a maximal ideal containing I.
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Notes

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References

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