integral domain
Appearance
English
[edit]Noun
[edit]integral domain (plural integral domains)
- (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 Integral domain on Wikipedia.Wikipedia
Synonyms
[edit]- (commutative ring in which the product of nonzero elements is nonzero): entire ring
Hypernyms
[edit]Hyponyms
[edit]Holonyms
[edit]Translations
[edit]nonzero commutative ring in which the product of nonzero elements is nonzero
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References
[edit]- ^ Jeff Miller, editor (2016), “Archived copy”, in Earliest Known Uses of Some of the Words of Mathematics[1], archived from the original on 17 August 2017
Further reading
[edit]- Zero-product property on Wikipedia.Wikipedia
- Dedekind–Hasse norm on Wikipedia.Wikipedia