integral domain en · NOUN
Meanings
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(qualifier:ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero.
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).
1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266:#91;#92;mathbb#123;Z#125;#59;#43;,#92;cdot#93;, #91;#92;mathbb#123;Z#125;#95;p,#43;#95;p,#92;times#95;p#93; with p a prime, #91;#92;mathbb#123;Q#125;#59;#43;,#92;cdot#93;, #91;#92;mathbb#123;R#125;#59;#43;,#92;cdot#93;, and #91;#92;mathbb#123;C#125;#59;#43;,#92;cdot#93; are all integral domains. The key example of an infinite integral domain is #91;#92;mathbb#123;Z#125;#59;#43;,#92;cdot#93;. In fact, it is from #92;mathbb#123;Z#125; that the term integral domain is derived. Our main example of a finite integral domain is #91;#92;mathbb#123;Z#125;#95;p,#43;#95;p,#92;times#95;p#93;, when p is prime.
2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171: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.
2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95:A ring R is an integral domain if and only if the polynomial ring R#91;x#93; is an integral domain.
For any integral domain there can be derived an associated field of fractions.
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| integral domains | Number=Plur | lexicographic |
Holonyms
Hypernyms
Hyponyms
Noetherian domain · principal ideal domain · unique factorization domain · Euclidean domain · field
Synonyms
entire ring (commutative ring in which the product of nonzero elements is nonzero)
Translations (11)
sh oblast celih (nonzero commutative ring in which the product of nonzero elements is nonzero) · sh oblast cijelih (nonzero commutative ring in which the product of nonzero elements is nonzero) · fi kokonaisalue (nonzero commutative ring in which the product of nonzero elements is nonzero) · sh domen (nonzero commutative ring in which the product of nonzero elements is nonzero) · sh домен (nonzero commutative ring in which the product of nonzero elements is nonzero) · fr anneau d'intégrité (nonzero commutative ring in which the product of nonzero elements is nonzero) · sh област целих (nonzero commutative ring in which the product of nonzero elements is nonzero) · it dominio d'integrità (nonzero commutative ring in which the product of nonzero elements is nonzero) · es dominio de integridad (nonzero commutative ring in which the product of nonzero elements is nonzero) · pl dziedzina całkowitości (nonzero commutative ring in which the product of nonzero elements is nonzero) · sh област цијелих (nonzero commutative ring in which the product of nonzero elements is nonzero)