Math 2.0

Commutative algebra Proposition

Cancellation in an integral domain

Builds on Integral domain

Statement

Let RR be an integral domain and let a,b,c∈Ra, b, c \in R with a≠0a \neq 0. If ab=aca b = a c, then b=cb = c.

Remarks

The proof is short: ab=aca b = a c gives a(b−c)=0a (b - c) = 0, and a≠0a \neq 0 gives b−c=0b - c = 0. The converse also holds: a nonzero commutative ring with this cancellation property is an integral domain. Cancellation fails in ℤ/6ℤ\mathbb{Z}/6\mathbb{Z}: 2⋅1=2⋅42 \cdot 1 = 2 \cdot 4, but 1≠41 \neq 4.

Reading path

To understand this page from the ground up, read these 6 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 ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionCancellation in an integral domain, statement, no evidence levelCancellation in an in…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. Zero divisor definition
  5. Integral domain definition
  6. Cancellation in an integral domain statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral_domain, Properties
About this page
Tag0GJM24EXP9
URIm2://0GJM24EXP9
Kindstatement, proposition
WritingW0 stub
MSC13G05
StatusThis copy of the site has no build data. Run m2 status 0GJM24EXP9.
UsesIntegral domain

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