On regular semigroup rings (Q5916460)

Let S be the free inverse semigroup of countable rank in the inverse semigroup variety \([x^ 2=x^ 3]\); evidently S is periodic. Let F be a field with char F\(=0\). Using a theorem by \textit{E. I. Kleiman} [Two theorems on combinatorial inverse semigroup varieties (Preprint 1980, 12 p., in Russian)] the author shows that S is not locally finite. Hence F[S] cannot be von Neumann regular. Simple semigroup observations show that F[H] is von Neumann regular for any principal factor H of S. That gives a negative answer to a question (Q.1) posed by \textit{J. Okniński} [Proc. R. Soc. Edinb., Sect. A 99, 145-151 (1984; Zbl 0564.20046)].