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]

   A ring $R$ is an integral domain if and only if the polynomial ring $R[x]$ 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:

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

Usage notes

For a list of several equivalent definitions, see Integral domain 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

Further reading

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