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.
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.