Factorization in monoid domains

In this work, we study several factorization properties in a monoid domain which are weaker than unique factorization.

In section 2, we investigate preliminary results of factorization in integral domains. Also we introduce several classes of monoids which are closely related to factorization in monoids. Next we study basic properties of factorizations in monoid domains and graded integral domains.

In section 3, we determine necessary and sufficient conditions for the group ring D[G] to be a BFD (resp., an FFD, an SFFD). Also we give necessary and sufficient conditions for the monoid domain D[S] to be a BFD (resp., an FFD, an SFFD) under some hypotheses. At the end of this section we characterize when the monoid domain D[S] is a UFD in terms of 2-factoriality.

In section 4, we show that any nonzero divisor class of a Krull domain D[S], where D is a UFD, contains a height-one prime ideal of D[S]. As a consequence, we can determine the elasticity of factorizations in the Krull monoid domain D[S].

In the final section, we first characterize the various factorization properties in generalized Rees rings. Next we compute the elasticity of factorizations in subrings of D[X] or D[[X]]. Finally we determine (necessary and) sufficient conditions for an A + XB[X] domain to be an HFD.