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 [6] 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, [9] 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 [1], 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.