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.
References in periodicals archive ?
The ring in this case is not an integral domain, since the modulus is composite.
5), for most implementations the lack of an integral domain here does not matter.
This section introduces, by analogy to integer division, a generalized polynomial division scheme which is nearly as fast, permits use of any polynomial, and may be computed over an integral domain.
The fact that the radix must be a zero divisor precludes R being an integral domain.
An integral domain is a commutative ring that possesses no zero divisors; that is, the multiplication operation is commutative, and the multiplication of two nonzero elements of the ring produces a nonzero result.
If r is composite, then R[prime] = R/r cannot be an integral domain.
To avoid such key biases being reflected in the hash value distribution, it is sufficient to insist that r be irreducible, that is, that R[prime] be an integral domain.
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.
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.
Other topics include polynomials, factorization in integral domains, p-groups and the Sylow theorems, Galois theory, and finiteness conditions for rings and modules.

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