integral domain
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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 pseudoDedekind 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#Definition 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

References[edit]
 ^ 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]
 Zeroproduct property on Wikipedia.Wikipedia
 Dedekind–Hasse norm on Wikipedia.Wikipedia