Math 2.0

Commutative algebra Corollary

Maximal ideals are prime

Builds on Maximal idealPrime idealR/M is a field iff M is maximalR/P is an integral domain iff P is prime

Statement

Let RR be a commutative ring and let MM be a maximal ideal of RR. Then MM is a prime ideal.

Remarks

The proof: R/MR/M is a field, so it is an integral domain, so MM is prime. The converse is false: the zero ideal of ℤ\mathbb{Z} is prime but not maximal. In ℤ\mathbb{Z}, every nonzero prime ideal is maximal.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionIdeal, definition, no evidence levelIdealdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionPrime ideal, definition, no evidence levelPrime idealdefinitionField, definition, no evidence levelFielddefinitionMaximal ideal, definition, no evidence levelMaximal idealdefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionThe only ideals of a field are zero and the whole field, statement, no evidence levelThe only ideals of a …statementQuotient ring, definition, no evidence levelQuotient ringdefinitionCorrespondence theorem for rings, statement, no evidence levelCorrespondence theore…statementR/P is an integral domain iff P is prime, statement, no evidence levelR/P is an integral do…statementR/M is a field iff M is maximal, statement, no evidence levelR/M is a field iff M …statementMaximal ideals are prime, statement, no evidence levelMaximal ideals are pr…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. Commutative ring definition
  4. Ideal definition
  5. Unit of a ring definition
  6. Zero divisor definition
  7. Prime ideal definition
  8. Field definition
  9. Maximal ideal definition
  10. Two-sided ideal definition
  11. Integral domain definition
  12. The only ideals of a field are zero and the whole field statement
  13. Quotient ring definition
  14. Correspondence theorem for rings statement
  15. R/P is an integral domain iff P is prime statement
  16. R/M is a field iff M is maximal statement
  17. Maximal ideals are prime statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Maximal_ideal, Properties
About this page
Tag05WAF5T0GZ
URIm2://05WAF5T0GZ
Kindstatement, corollary
WritingW0 stub
MSC13A15
StatusThis copy of the site has no build data. Run m2 status 05WAF5T0GZ.
UsesMaximal ideal, Prime ideal, R/M is a field iff M is maximal, R/P is an integral domain iff P is prime

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