# principal ideal

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## principal ideal

[′prin·sə·pəl ī′dēl]
(mathematics)
The smallest ideal of a ring which contains a given element of the ring.
References in periodicals archive ?
An integral domain is a principal ideal domain (PID) if every prime ideal is principal.
Let R be a principal ideal domain with an identity element and let [R.sup.mxn] denote the set of m x n matrices over R.
These results on Euclidean self-dual cyclic codes have been generalized to abelian codes in group algebras  and the complete characterization and enumeration of Euclidean self-dual abelian codes in principal ideal group algebras (PIGAs) have been established.
Let 1 be the prime ideal of F' dividing l, and [lambda] an algebraic integer in F' generating the principal ideal 1.
The principal ideal generated by 1 [direct sum] [epsilon] is a natural choice for our ideal I.
A principal ideal (resp., principal filter) is a set of the form [down arrow] x = [{y [member of] L: y [less than or equal to] x} (resp., [up arrow] x = [{y [member of] L: x [less than or equal to] y}).
In particular,  deals with the factorization of formal power series over principal ideal domains.
The subsemigroup ([N'.sub.0], *) is a proper and principal ideal of (H', *).
In , Campoli proved that Z[[theta]] is a principal ideal domain which is not a Euclidean domain.
Using mainly concrete constructions, Gerstein gives a brief introduction to classical forms, then moves to quadratic spaces and lattices, valuations, local fields, p-adic numbers, quadratic spaces over Qp and over Q, lattices over principal ideal domains, initial integral results, the local-global approach to lattices, and applications to cryptography.
Since ker P is an ideal in C[z] and since every ideal in C[z] is a principal ideal (cf.
The two functions [Mathematical Expression Omitted] generate the same principal ideal if and only if v([f.sub.i]) = v([g.sub.i]) for every valuation v as above.

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