Rings With a Finite Number of Orbits Under the Regular Action

Abstract

Abstract. Let R be a ring with identity, X(R) the set of all nonzero, non-units of R and G(R) the group of all units of R. We show that for a matrix ring Mn(D), n ≥ 2, if a, b are singular matrices of the same rank, then |o ℓ (a)| = |o ℓ (b)|, where o ℓ (a) and o ℓ (b) are the orbits of a and b, respectively, under the left regular action. We also show that for a semisimple Artinian ring R such that, with D i infinite division rings of the same cardinalities or R is isomorphic to the ring of 2 × 2 matrices over a finite field if and only if |o ℓ (x)| = |o ℓ (y)| for all x, y ∈ X(R).