On rings with finite number of orbits. (Q2438276)

Let \(U(R)\) be the group of units of a ring \(R\). Let \(S\) be either of the following sets: the set of elements of \(R\), the set of left ideals of \(R\), the set of principal left ideals of \(R\), the set of two-sided ideals of \(R\). Then the group \(U(R)\times U(R)\) acts on the set \(S\) by left and right multiplication. This paper describes various properties of rings \(R\) with a finite number of orbits under the action of group \(U(R)\times U(R)\) on set \(S\).