On monoids, $2$-firs, and semifirs

Abstract

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 recover Cedó’s results for both constructions. Any homomorphism from a monoid $M$ into $\mathbb {Z}$ induces a $\mathbb {Z}$-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 $\mathbb {Z}$-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 $DM$ is a $2$-fir, 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 of infinite cyclic monoids. Several open questions are noted.