Journal of the
Korean Mathematical Society JKMS

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 $|o_{\ell}(a)| = |o_{\ell}(b)|$, 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 $X(R) \neq \emptyset$, $R \cong \oplus_{i=1}^{m} M_{n_{i}}(D_{i})$, with $D_{i}$ infinite division rings of the same cardinalities or $R$ is isomorphic to the ring of $2 \times 2$ matrices over a finite field if and only if $|o_{\ell}(x)| = |o_{\ell}(y)|$ for all $x, y \in X(R)$.

Keywords: left (right) regular action, orbit, left Artinian ring

MSC numbers: Primary 16W22; Secondary 16P20