Math 2.0

Commutative algebra Basic notions Theorem

R/P is an integral domain iff P is prime

Builds on Quotient ringPrime idealIntegral domain

Statement

Let RR be a commutative ring and let PP be an ideal of RR. Then the quotient ring R/PR/P is an integral domain if and only if PP is a prime ideal.

Remarks

The condition 1+P≠0+P1 + P \neq 0 + P in R/PR/P is the condition P≠RP \neq R. The condition that (a+P)(b+P)=0(a + P)(b + P) = 0 implies a+P=0a + P = 0 or b+P=0b + P = 0 is the condition that ab∈Pa b \in P implies a∈Pa \in P or b∈Pb \in P. For example, ℤ/nℤ\mathbb{Z}/n\mathbb{Z} is an integral domain exactly when nn is prime or n=0n = 0.

Reading path

To understand this page from the ground up, read these 10 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 levelIdealdefinitionZero divisor, definition, no evidence levelZero divisordefinitionPrime ideal, definition, no evidence levelPrime idealdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionQuotient ring, definition, no evidence levelQuotient ringdefinitionR/P is an integral domain iff P is prime, statement, no evidence levelR/P is an integral do…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. Zero divisor definition
  6. Prime ideal definition
  7. Integral domain definition
  8. Two-sided ideal definition
  9. Quotient ring definition
  10. R/P is an integral domain iff P is prime statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Prime_ideal, Properties
About this page
Tag0AVXBEC05W
URIm2://0AVXBEC05W
Kindstatement, theorem
WritingW0 stub
MSC13A15
StatusThis copy of the site has no build data. Run m2 status 0AVXBEC05W.
UsesQuotient ring, Prime ideal, Integral domain

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