UC Berkeley

Several authors have studied the question of when the monoid ring DM of a monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir, or a 2-fir (definitions recalled in §1). It is known that for M nontrivial, a necessary condition for any of these properties to hold is that D be a division ring. Under that assumption, necessary and sufficient conditions on M are known for DM to be a right or left fir, and various conditions on M have been proved necessary or sufficient for DM to be a 2-fir or semifir. A sufficient condition for DM to be a semifir is that M be a direct limit of monoids which are free products of free monoids and free groups. Warren Dicks has conjectured that this is also necessary. However F. Cedó has given an example of a monoid M which is not such a direct limit, but satisfies all the known necessary conditions for DM to be a semifir. It is an open question whether for this M, the rings DM are semifirs.We note here some reformulations of the known necessary conditions for a monoid ring DM to be a 2-fir or a semifir, motivate Cedó’s construction and a variant thereof, and recoverCedó’s results for both constructions. Any homomorphism from a monoid M into Z induces a Z-grading on DM, and we show that for the two monoids just mentioned, the rings DM are “homogeneous semifirs” with respect to all such nontrivial Z-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free of unique rank. If M is a monoid such that DM is an n-fir, and N a “well-behaved” submonoid of M, we prove some properties of the ring DN. Using these, we show that for M a monoid such that DM.