英语词汇“integral domain”的英英意思、用法、释义、翻译、读音、例句、短语、词组-英语词典

d. Mathematics. An algebraic system with two binary operations defined by postulates stronger than those for a ring but weaker than those for a field; esp. (more fully integral domain), a commutative ring in which the cancellation law holds for multiplication of non-zero elements and (with most writers) which has a unit element for multiplication.

ΚΠ

1896 Bull. Amer. Math. Soc. 3 102 The idea of the adjunction of its root[s]..to the totality of numbers heretofore in use, viz., the rational numbers, is now developed, and the properties of rational functions of the elements of this new ‘domain’..are taken up.

1904 F. Cajori Introd. Mod. Theory Equations xiii. 134 A set of numbers is called a domain of rationality or simply a domain, when the sums, differences, products, and quotients of any numbers in the set (excluding only the quotients obtained through division by 0) always yield as results numbers belonging to the set. All rational numbers..constitute such a domain.

1937 A. A. Albert Mod. Higher Algebra ii. 27 The most important type of integral domain is the field.

1941 G. Birkhoff & S. MacLane Surv. Mod. Algebra iv. 97 By a unique factorization domain (sometimes called a ‘Gaussian domain’) is meant an integral domain in which (i) any element not a unit can be factored into primes, (ii) this factorization is unique to within order and unit factors.

1958 O. Zariski & P. Samuel Commutative Algebra I. i. 22 An important class of unique factorization domains is given by the so-called euclidean domains or rings admitting a division algorithm.

1965 J. J. Rotman Theory of Groups iv. 66 A domain is a commutative ring with unit that has no divisors of zero.

单词	integral domain
释义	> as lemmas integral domain d. Mathematics. An algebraic system with two binary operations defined by postulates stronger than those for a ring but weaker than those for a field; esp. (more fully integral domain), a commutative ring in which the cancellation law holds for multiplication of non-zero elements and (with most writers) which has a unit element for multiplication. ΚΠ 1896 Bull. Amer. Math. Soc. 3 102 The idea of the adjunction of its root[s]..to the totality of numbers heretofore in use, viz., the rational numbers, is now developed, and the properties of rational functions of the elements of this new ‘domain’..are taken up. 1904 F. Cajori Introd. Mod. Theory Equations xiii. 134 A set of numbers is called a domain of rationality or simply a domain, when the sums, differences, products, and quotients of any numbers in the set (excluding only the quotients obtained through division by 0) always yield as results numbers belonging to the set. All rational numbers..constitute such a domain. 1937 A. A. Albert Mod. Higher Algebra ii. 27 The most important type of integral domain is the field. 1941 G. Birkhoff & S. MacLane Surv. Mod. Algebra iv. 97 By a unique factorization domain (sometimes called a ‘Gaussian domain’) is meant an integral domain in which (i) any element not a unit can be factored into primes, (ii) this factorization is unique to within order and unit factors. 1958 O. Zariski & P. Samuel Commutative Algebra I. i. 22 An important class of unique factorization domains is given by the so-called euclidean domains or rings admitting a division algorithm. 1965 J. J. Rotman Theory of Groups iv. 66 A domain is a commutative ring with unit that has no divisors of zero. extracted from domainn. < as lemmas
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