Monoid rings that are firs (Q919082)

Let M be a free product of a free group and a free monoid. Then the monoid ring KM over any skew field K is a fir. The author cites five distinct proofs of this result [cf. e.g. \textit{P. M. Cohn}, Free rings and their relations (2nd ed., Academic Press, London 1985; Zbl 0659.16001)] and gives another very direct proof using transductions, which produces a basis consisting of monomials for each left ideal.