Math 2.0

Commutative algebra Basic notions Theorem

R/M is a field iff M is maximal

Builds on Quotient ringMaximal idealFieldCorrespondence theorem for ringsThe only ideals of a field are zero and the whole field

Statement

Let RR be a commutative ring and let MM be an ideal of RR. Then the quotient ring R/MR/M is a field if and only if MM is a maximal ideal.

Remarks

The proof uses the correspondence theorem: the ideals of R/MR/M are the images of the ideals of RR that contain MM. So MM is maximal if and only if R/MR/M is nonzero and has only the two ideals {0}\{0\} and R/MR/M. A nonzero commutative ring with only these two ideals is a field, by the converse for fields. For example, ℤ/pℤ\mathbb{Z}/p\mathbb{Z} is a field because pℤp\mathbb{Z} is maximal for pp prime.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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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 ringdefinitionField, definition, no evidence levelFielddefinitionMaximal ideal, definition, no evidence levelMaximal idealdefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionThe 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/M is a field iff M is maximal, statement, no evidence levelR/M is a field iff M …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. Field definition
  7. Maximal ideal definition
  8. Two-sided ideal definition
  9. The only ideals of a field are zero and the whole field statement
  10. Quotient ring definition
  11. Correspondence theorem for rings statement
  12. R/M is a field iff M is maximal statement

Where this leads

23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Maximal_ideal, Properties
About this page
Tag0WD3SE7S1X
URIm2://0WD3SE7S1X
Kindstatement, theorem
WritingW0 stub
MSC13A15
StatusThis copy of the site has no build data. Run m2 status 0WD3SE7S1X.
UsesQuotient ring, Maximal ideal, Field, Correspondence theorem for rings, The only ideals of a field are zero and the whole field

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