# 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 ?
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.
In particular,  deals with the factorization of formal power series over principal ideal domains.
Since S is a semilattice, every principal ideal SsS has a unique generator s -[member of]S, so this order is well defined.
0] is a principal ideal with countable infinite generators.
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.
Let I be a coherent ideal of [Mathematical Expression Omitted], that is a principal ideal generated by a function [Mathematical Expression Omitted].
This edition, which includes new sections on modules(free, semisimple and projective), modules over principal ideal domains, semidirect products, and the Wedderburn-Artin theorem also includes new appendices on Zorn's lemma and the proof of the recursive theorem.
Rather, we advocate patience and prudence as principal ideals for the road to recovery.

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