On monoids, 2-firs, and semifirs

TL;DR

This paper investigates conditions under which monoid rings are free ideal rings, semifirs, or 2-firs, exploring known results, conjectures, and new constructions, including grading properties and submonoid implications.

Contribution

It provides new insights into the structure of monoid rings, examines Cedó's example challenging existing conjectures, and explores grading and submonoid properties related to fir conditions.

Findings

Cedó's monoid satisfies known necessary conditions but remains an open case for being a semifir.

Monoid rings that are 2-firs have elements forming equivalence classes with specific algebraic structures.

Homogeneous semifir properties are established for certain monoids with respect to all nontrivial -gradings.

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 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. W.Dicks has conjectured that this is also necessary. However F.Ced\'o has given an example of a monoid M which is not such a direct limit, but satisfies 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 some reformulations of the known necessary conditions for DM to be a 2-fir or a semifir, motivate Ced\'o's construction and a variant, and recover Ced\'o'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 \Z-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 of M, we obtain results on 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 infinite cyclic monoids. Several open questions are noted.