Artinian groupoid rings (Q1386725)

Let \(F\) be a field, let \(G\) be a groupoid and let \(FG\) be a corresponding groupoid algebra. The author proves the following interesting theorems. There exists a groupoid \(H\) containing \(G\) as a subgroupoid and such that \(FH\) has three right ideals only. Let \(F\) be the field of rational numbers, and let \(G\) be cancellative; then there exists a quasigroup \(Q\) containing \(G\) as a subgroupoid and such that \(FQ\) has three right ideals only. These theorems show that Zelmanov's well known theorem on artinian semigroup rings cannot be extended in general to groupoid rings.