In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and more specifically in
ring theory, Krull's theorem, named after
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
, asserts that a
nonzero ring
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
has at least one
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
. The theorem was proved in 1929 by Krull, who used
transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for a ...
. The theorem admits a
simple proof using Zorn's lemma, and in fact is equivalent to
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
,
which in turn is equivalent to the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
.
Variants
* For
noncommutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s, the analogues for maximal left ideals and maximal right ideals also hold.
* For
pseudo-rings, the theorem holds for
regular ideal In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.
In operator theory, a right ideal (ring theory), ideal \mathfrak in a (possibly) non-unital ring ''A'' is said to be regular (or modular) if there exists an ele ...
s.
* 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
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An '' additive id ...
(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''.
Notes
References
*
*{{cite journal , first=W. , last=Hodges , title=Krull implies Zorn , journal=
Journal of the London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
, volume=s2-19 , issue=2 , year=1979 , pages=285–287 , doi=10.1112/jlms/s2-19.2.285
Ideals (ring theory)
Theorems in algebra