integral domain

Noun

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Noun

**integral domain** (*plural* integral domains)

- (
*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).

- For
**2013**, Marco Fontana, Evan Houston, Thomas Lucas,*Factoring Ideals in*, Springer, page 95 ↗,**Integral Domains**- 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.

- An
**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.

- [\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

- (
*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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