Math 2.0

Commutative algebra Proof

Proof that maximal ideals are prime

Builds on Commutative ringMaximal idealPrime idealR/M is a field iff M is maximalFieldIntegral domainR/P is an integral domain iff P is prime

Proof

  1. Let RR be a commutative ring and let MM be a maximal ideal of RR.
  2. Since MM is maximal, the quotient ring R/MR/M is a field.
  3. Every field KK is an integral domain. It is commutative with 1≠01 \neq 0. If xy=0x y = 0 in KK and x≠0x \neq 0, then y=(x−1x)y=x−1(xy)=x−1⋅0=0y = (x^{-1} x) y = x^{-1} (x y) = x^{-1} \cdot 0 = 0.
  4. So R/MR/M is an integral domain.
  5. Since R/MR/M is an integral domain, MM is a prime ideal.

Idea

Maximal ideals are exactly those with a field as quotient, and prime ideals are exactly those with an integral domain as quotient. Since every field is an integral domain, every maximal ideal is prime.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first. The path takes the proofs Proof that R/P is an integral domain iff P is prime, Proof of the correspondence theorem for rings, Proof that R/M is a field iff M is maximal, Proof that the only ideals of a field are zero and the whole field.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionGroup, concept, no evidence levelGroupconceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionRing, definition, no evidence levelRingdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionCommutative 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 divisordefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionPrime ideal, definition, no evidence levelPrime idealdefinitionField, definition, no evidence levelFielddefinitionMaximal ideal, definition, no evidence levelMaximal idealdefinitionSubset, definition, no evidence levelSubsetdefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionProof that the only ideals of a field are zero and the whole field, proof, no evidence levelProof that the only i…proofQuotient ring, definition, no evidence levelQuotient ringdefinitionThe only ideals of a field are zero and the whole field, statement, no evidence levelThe only ideals of a …statementProof that R/M is a field iff M is maximal, proof, no evidence levelProof that R/M is a f…proofProof that R/P is an integral domain iff P is prime, proof, no evidence levelProof that R/P is an …proofFunction, definition, no evidence levelFunctiondefinitionR/P is an integral domain iff P is prime, statement, no evidence levelR/P is an integral do…statementCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionImage of a set, definition, no evidence levelImage of a setdefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionProof of the correspondence theorem for rings, proof, no evidence levelProof of the correspo…proofCorrespondence theorem for rings, statement, no evidence levelCorrespondence theore…statementR/M is a field iff M is maximal, statement, no evidence levelR/M is a field iff M …statementProof that maximal ideals are prime, proof, no evidence levelProof that maximal id…proof
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. Element definition
  2. Group concept
  3. Proposition concept
  4. Set definition
  5. Ring definition
  6. Implication definition
  7. Predicate definition
  8. Ordered pair definition
  9. Commutative ring definition
  10. Ideal definition
  11. Unit of a ring definition
  12. Zero divisor definition
  13. Existential quantifier definition
  14. Universal quantifier definition
  15. Cartesian product definition
  16. Prime ideal definition
  17. Field definition
  18. Maximal ideal definition
  19. Subset definition
  20. Two-sided ideal definition
  21. Integral domain definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Proof that the only ideals of a field are zero and the whole field proof
  25. Quotient ring definition
  26. The only ideals of a field are zero and the whole field statement
  27. Proof that R/M is a field iff M is maximal proof
  28. Proof that R/P is an integral domain iff P is prime proof
  29. Function definition
  30. R/P is an integral domain iff P is prime statement
  31. Codomain definition
  32. Injective function definition
  33. Image of a set definition
  34. Preimage of a set definition
  35. Surjective function definition
  36. Bijective function definition
  37. Proof of the correspondence theorem for rings proof
  38. Correspondence theorem for rings statement
  39. R/M is a field iff M is maximal statement
  40. Proof that maximal ideals are prime proof

Where this leads

28 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Maximal_Ideal_of_Commutative_and_Unitary_Ring_is_Prime_Ideal
About this page
TagA1QC3J125A
URIm2://A1QC3J125A
Kindproof
WritingW0 stub
MSC13A15
ProvesMaximal ideals are prime
UsesCommutative ring, Maximal ideal, Prime ideal, R/M is a field iff M is maximal, Field, Integral domain, R/P is an integral domain iff P is prime
Changed4 November 2026

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