Math 2.0

Commutative algebra Definition

Integral domain

Builds on Commutative ringZero divisor

Definition

An integral domain is a commutative ring RR with these two properties.

  1. 1≠01 \neq 0 in RR.
  2. For all a,b∈Ra, b \in R, if ab=0a b = 0, then a=0a = 0 or b=0b = 0.

Equivalently, RR is a nonzero commutative ring in which 00 is the only zero divisor.

Remarks

The integers ℤ\mathbb{Z} and every field are integral domains. The ring ℤ/6ℤ\mathbb{Z}/6\mathbb{Z} is not an integral domain, because 2⋅3=02 \cdot 3 = 0. In an integral domain, cancellation holds for nonzero factors. Some authors, and Mathlib, also use the word domain for a noncommutative ring with property 2.

Reading path

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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 domaindefinition
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.
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Zero divisor definition
  5. Integral domain definition

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144 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral_domain, Definition
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WritingW1 stated
MSC13G05
DefinesIntegral domain
UsesCommutative ring, Zero divisor

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