Math 2.0

Commutative algebra Basic notions Theorem

Existence of maximal ideals

Builds on Maximal ideal

Statement

Let RR be a ring and let II be an ideal of RR with I≠RI \neq R. Then there is a maximal ideal MM of RR with I⊆MI \subseteq M. In particular, every nonzero ring has a maximal ideal.

Remarks

The proof uses Zorn's lemma. The union of a chain of proper ideals that contain II is again a proper ideal, because it does not contain 11. The statement for every nonzero ring is equivalent to the axiom of choice, over the other axioms of ZF set theory. The statement holds for left, right and two-sided ideals.

History

Wolfgang Krull proved the theorem in 1929 with transfinite induction.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionIdeal, definition, no evidence levelIdealdefinitionMaximal ideal, definition, no evidence levelMaximal idealdefinitionExistence of maximal ideals, statement, no evidence levelExistence of maximal …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Ideal definition
  4. Maximal ideal definition
  5. Existence of maximal ideals statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Krull%27s_theorem, lead section
About this page
Tag02HJYT53F7
URIm2://02HJYT53F7
Aliaseskrull-s-theorem
Kindstatement, theorem
WritingW0 stub
MSC13A15
StatusThis copy of the site has no build data. Run m2 status 02HJYT53F7.
UsesMaximal ideal
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