Math 2.0

statement corollary

Maximal ideals are 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.

Explanations

L1 human

Every field is an integral domain. So when R/MR/M is a field, R/MR/M is an integral domain, and MM is prime. The zero ideal of the integers shows that a prime ideal need not be maximal.

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To read this node, read these 17 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Maximal_ideal, Properties

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