integral domain

integral domain

English

Wikibooks

Noun

integral domain (plural integral domains)

1. (algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero. [from 1911] [1]

   A ring ${\\displaystyle R}$ is an integral domain if and only if the polynomial ring ${\\displaystyle R[x]}$ is an integral domain.

   For any integral domain there can be derived an associated field of fractions.

   • 1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266,

     For integral domains, we will use a^-1 to designate the multiplicative inverse of a (if it has one; since not all elements need have inverses, this notation can be used only where it can be shown that an inverse exists).

   • 2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95,

     An integral domain is said to have strong pseudo-Dedekind factorization if each proper ideal can be factored as the product of an invertible ideal (possibly equal to the ring) and a finite product of pairwise comaximal prime ideals with at least one prime in the product.

   • 2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171,

     ${\\displaystyle [\\mathbb {Z}$ ;+,\\cdot ]} , ${\\displaystyle [\\mathbb {Z} _{p},+_{p},\\times _{p}]}$ with ${\\displaystyle p}$ a prime, ${\\displaystyle [\\mathbb {Q}$ ;+,\\cdot ]} , ${\\displaystyle [\\mathbb {R}$ ;+,\\cdot ]} , and ${\\displaystyle [\\mathbb {C}$ ;+,\\cdot ]} are all integral domains. The key example of an infinite integral domain is ${\\displaystyle [\\mathbb {Z}$ ;+,\\cdot ]} . In fact, it is from ${\\displaystyle \\mathbb {Z} }$ that the term integral domain is derived. Our main example of a finite integral domain is ${\\displaystyle [\\mathbb {Z} _{p},+_{p},\\times _{p}]}$, when ${\\displaystyle p}$ is prime.

Usage notes

For a list of several equivalent definitions, see Integral domain#Definition on Wikipedia.Wikipedia

Synonyms

• (commutative ring in which the product of nonzero elements is nonzero): entire ring

Hypernyms

• commutative ring
• domain

Hyponyms

• unique factorization domain, Noetherian domain
  • principal ideal domain
    • Euclidean domain
      • field

Holonyms

• field of fractions

Translations

nonzero commutative ring in which the product of nonzero elements is nonzero

Finnish: kokonaisalue French: anneau d'intégrité (fr) m Italian: dominio d'integrità m Serbo-Croatian: Cyrillic: област цијелих, област целих, домен Latin: oblast cijelih, oblast celih, domen (sh) Spanish: dominio de integridad m

References

1. Jeff Miller, editor (2016), “Archived copy”, in Earliest Known Uses of Some of the Words of Mathematics‎, archived from the original on 2017-08-17, retrieved 19 September 2017

Further reading

• Zero-product property on Wikipedia.Wikipedia
• Dedekind–Hasse norm on Wikipedia.Wikipedia