On group rings which are not Ore domains (Q1118667)

The paper is motivated by the problem of finding necessary and sufficient conditions on a semigroup S in order that the semigroup ring R[S] of S over a ring R would be an Ore ring. It is shown that the group ring R[G] is never Ore if G has a free subgroup of rank two. One of the tools used is the free rank of a finite subset A of G defined as \(fr(A)=\inf (| AB| /| B|)\) where B runs over all finite subsets of G. It is also proved that if \(n>1\) and \(A=\{1,a_ 1,...,a_ n\}\), then \(rk(A)=n\) if and only if \(a_ 1,...,a_ n\) freely generate a free subgroup of G.