integral domain


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integral domain

[′int·ə·grəl dō′mān]
(mathematics)
A commutative ring with identity where the product of nonzero elements is never zero. Also known as entire ring.
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References in periodicals archive ?
NQR(Z) the neutrosophic quadruple ring of integers is a neutrosophic quadruple integral domain.
On neutrosophic quadruple algebraic structures
In [6], it is shown that an integral domain with no universal side divisors can not be Euclidean.
Euclidean semimodules
The proof of this lemma uses (2, Theorem (2.10)) stating that given any integral domain R and a matrix X of variables, the ideal generated by the maximal minors of X in the polynomial ring R[X] is prime.
Colored Tutte polynomials and composite knots
(Conversely, a nearly limitless family of hash functions can be had, each member of which is identified with a different choice of T.) Second, the polynomial [x.sup.w] + 1 is composite, so that the ring is not an integral domain. As the Appendix points out (Section A.5), for most implementations the lack of an integral domain here does not matter.
Recursive hashing functions for n-grams
Chapters four through seven cover abstract groups and monoids, orthogonal groups, stochastic matrices, Lagrange's theorem, groups of units of monoids, homomorphisms, rings, and integral domains. The first seven chapters provide basic coverage of abstract algebra, suitable for a one-semester or two-quarter course.
Introduction to Abstract Algebra
This is followed by a range of current topics such as multilinear algebra, matrix polynomials and equations, perturbation theory, pseudospectra, inverse problems, integral domains, and spectral sets, among many others.
Handbook of Linear Algebra, 2nd Edition
Some simple considerations about symmetry properties of the integral domains T in (2)-(4) deserve to be better investigated.
On the divergenceless property of the magnetic induction field
This comprehensive treatment of an under-studied aspect of commutative algebra describes various simple extensions and their properties, in particular properties of simple ring extensions of Noetherian integral domains. Oda (mathematics, Kochi U.) and Yoshida (mathematics, Okayama U.
Simple extensions with the minimum degree relations of integral domains
They then discuss integers, rings and ordered integral domains. They also include an elementary proof of the Fundamental Theorem of Algebra.
Number Systems: An Introduction to Algebra and Analysis
Among the topics are greatest common divisors, integer multiples and exponents, quotients of polynomial rings, divisibility and factorization in integral domains, subgroups of cyclic groups, cosets and Lagrange's theorem, the fundamental theorem of finite abelian groups, and check digits.
Abstract Algebra: An Inquiry-Based Approach
Nicholson starts with a review of proofs, sets, mappings and equivalences, then in 11 chapters covers integers and permutations, groups, rings, polynomials, factorization in integral domains, fields, modules over principal ideal domains, p-groups and the Sylow theorems, series of subgroups, Galois theory and finiteness conditions for rings and modules.
Introduction to abstract algebra, 3d ed
Other topics include polynomials, factorization in integral domains, p-groups and the Sylow theorems, Galois theory, and finiteness conditions for rings and modules.
Introduction to abstract algebra, 4th ed

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