integral domain

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,, 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.
  • (commutative ring in which the product of nonzero elements is nonzero) entire ring
  • French: anneau d'intégrité
  • Italian: dominio d'integrità
  • Spanish: dominio de integridad

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