On monoids, 2-firs, and semifirs
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On monoids, 2-firs, and semifirs


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 section 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 of that construction, and recover Cedó'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 in question, the rings DM are "homogeneous semifirs" with respect to all such nontrivial gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free. If M is a monoid such that DM is an n-fir, and N a "well-behaved" submonoid N of M, we prove some properties of DN. Using these, we show that if M is any monoid having a monoid ring DM which is a 2-fir, then mutual commutativity is an equivalence relation on nonidentity elements of M, and each equivalence class, together with the identity element, is a directed union of infinite cyclic groups or infinite cyclic monoids. Several open questions are noted.

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