Construction of left fir monoid rings (Q1327064)

In any monoid \(M\) (always with cancellation) define a preordering by putting \(a\leq b\) if \(b=ad\) for some \(d\in M\); this defines a partial ordering of classes of right associated elements of \(M\). \textit{I. B. Kozhukhov} [Algebra Logika 21, 37-59 (1982; Zbl 0512.16004)] has shown that for any ring \(K\) the monoid ring \(KM\) is a left fir if and only if \(K\) is a skew field, \(M\) has cancellation and the above partial order is a well-ordering on the left factors of any element, the unit group \(U(M)\) of \(M\) is free and \(M\) is irreflexive i.e. \(b=abc\) for a non-unit \(b\) implies \(a=c=1\). Such an \(M\) is also called a left fir monoid. For any element \(c\) of \(M\) the set of left factors (up to right associates) forms an ordinal \(\tau(c)\) and if \(\tau(M)=\sup\{\tau(c)\mid c\in M\}\), Kozhukhov proposed an example, for any ordinal \(\alpha\), of a left fir monoid \(M\) with \(\tau(M)=\alpha\). However his construction contains a mistake. The authors make a careful analysis of left fir monoids which results in a precise description of left fir monoids \(M\) with a given \(\tau(M)\). In particular this leads to a construction of a left fir monoid \(M\) with a given free group as group of units and a given ordinal as value of \(\tau(M)\).