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 $M_n(D)$, $n{geq}2$, if a, b are singular matrices of the same rank, then ${mid}o_{ell}(a){mid}={mid}o_{ell}(b){mid}$, where $o_{ell}(a)$ and $o_{ell}(b)$ 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 $X(R){ eq}{emptyset}$, $$R{{sim_=}}{oplus}^m_{i=1}M_n_i(D_i)$$, with $D_i$ infinite division rings of the same cardinalities or R is isomorphic to the ring of $2{ imes}2$ matrices over a finite field if and only if ${mid}o_{ell}(x){mid}={mid}o_{ell}(y){mid}$ for all $x,y{in}X(R)$.