On finite factorization Puiseux algebras

An integral domain D is called a finite factorization domain (FFD) if every nonzero nonunit element of D has only finitely many non-associate divisors. In 1998, for an integral domain D and a cancellative torsion-free monoid S such that each nonzero element of its quotient group is of type (0,0,…), Kim proved that the monoid domain D[S] is an FFD if and only if D is an FFD and S is an FFM. However, it is still open whether a monoid algebra K[S] is an FFD provided that S is a reduced FFM. In this paper, we show that a Puiseux algebra K[S] is an FFD if and only if S is an FFM, when K is a finitely generated field of characteristic 0. This would provide a large class of one-dimensional monoid algebras with finite factorization property. We also prove that every generalized cyclotomic polynomial has the finite factorization property in K[S] where S is a reduced FFM and K is an arbitrary field of characteristic 0.