An algorithm for commutative semigroup algebras which are principal ideal rings

Associative and commutative algebras with identity have various well-known applications. In particular, many classical codes are ideals in commutative algebras (see [4], [12] for references). Computer storage, encoding and decoding algorithms simplify if all these codes have single generator polynomials. Thus it is of interest to determine when all ideals of an algebra are principal. In [5] Decruyenaere, Jespers and Wauters characterized commutative semigroup algebras with identity which are principal ideal rings. In the more general case of noncommutative algebras all principal ideal semigroup algebras have been described by Jespers and Okninski [9]. For preliminaries on semigroup rings and other related constructions we refer to [13] and [10]. In this paper we develop an algorithm which, given a presentation for a commutative semigroup S and the characteristic of a field k, decides whether the semigroup algebra k[S] is a principal ideal ring with identity. This builds