On the algebra of a free inverse monoid (Q1923957)

Let \(S\) be an ideal of a free inverse monoid \(M\). It is shown that for the semigroup algebra \(F[S]\) over a field \(F\) the following three conditions are equivalent: (i) \(F[S]\) is prime; (ii) \(F[S]\) is primitive, (iii) \(M\) has infinite rank. This is an extension of a theorem of \textit{P. V. Silva} [Proc. R. Soc. Edinb., Sect. A 120, No. 3/4, 191-197 (1992; Zbl 0780.20043)] and it complements the result of the second author showing that \(F[M]\) is never prime if \(M\) is of finite rank [ibid. 107, 175-196 (1987; Zbl 0627.20041)].