Von Neumann finite endomorphism rings. (Q1433046)

A ring is called von Neumann finite if every left inverse in the ring is also a right inverse. Let \(V\) be a vector space over a field \(K\), then it is easy to see that the endomorphism ring \(\text{End}_K(V)\) is von Neumann finite if and only if \(V\) is finite dimensional. If \(G\) is a group, Kaplansky conjectured that the group ring of \(G\) over \(K\) is von Neumann finite. Now let \(K\) be a field of characteristic zero, \(G\) a group acting on a nonempty set \(X\) and \(KX\) the permutation module induced by this action. The author proves that the endomorphism ring \(\text{End}_{K[G]}(KX)\) is von Neumann finite under certain conditions on the action of \(G\) on \(X\). This generalizes a classical result of Kaplansky for the group ring of \(G\) over \(K\).