integral domain

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English[edit]

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Noun[edit]

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 is an integral domain if and only if the polynomial ring 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,
      , with a prime, , , and are all integral domains. The key example of an infinite integral domain is . In fact, it is from that the term integral domain is derived. Our main example of a finite integral domain is , when is prime.

Usage notes[edit]

For a list of several equivalent definitions, see Wikipedia-logo-v2.svg Integral domain#Definition on Wikipedia.Wikipedia

Synonyms[edit]

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

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Translations[edit]

References[edit]

  1. ^ Jeff Miller, editor (2016) Earliest Known Uses of Some of the Words of Mathematics[1], archived from the original on 17 August 2017

Further reading[edit]