Definitions for integral domain
in·te·gral do·main

WiktionaryRate this definition:0.0 / 0 votes

1. integral domainnoun

   A commutative ring with identity not equal to zero which has no zero divisors.

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1. Integral domain

   In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c. "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings.

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1. Integral domain

   In mathematics, and specifically in abstract algebra, an integral domain is a commutative ring without zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. The above is how "integral domain" is almost universally defined, but there is some variation. Some authors do not require a unit element. Noncommutative integral domains are sometimes admitted. However, this article follows the much more usual convention of reserving the term integral domain for the commutative case and using domain for the noncommutative case. Some sources, notably Lang, use the term entire ring for integral domain. Some specific kinds of integral domains are given with the following chain of class inclusions: The absence of zero divisors means that, in an integral domain, the cancellation property holds for multiplication by any nonzero element a: an equality ab = ac implies b = c.

How to pronounce integral domain?

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Numerology

1. Chaldean Numerology

   The numerical value of integral domain in Chaldean Numerology is: 1

2. Pythagorean Numerology

   The numerical value of integral domain in Pythagorean Numerology is: 7

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Integral_Domain
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Integral_Domain
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=integral domain