Rings with a finite number of orbits under the regular action. (Q2877773)

Throughout the paper under review, by a ring the authors mean an associative ring with unit. For a ring \(R\), they denote by \(G(R)\) and \(X(R)\) the unit group of \(R\) and the set of nonzero nonunits of \(R\), respectively. In the paper, the authors concentrate on the natural group action of \(G(R)\) on \(X(R)\) by left multiplication called the left regular action. For every \(x\in X(R)\), the orbit of \(x\) under the left regular action of \(G(R)\) on \(X(R)\) is denoted by \(o_l(x)\).NEWLINENEWLINE In Section 2, the authors generalize [\textit{J.-A. Cohen} and \textit{K. Koh}, J. Pure Appl. Algebra 60, No. 2, 139-153 (1989; Zbl 0692.16024)]. They prove that (Theorem 2.3) for \(R=M_n(F)\), the ring of \(n\times n\) matrices over a finite field \(F\) with \(n\geq 2\), if \(\text{rank}(x)=s\) for some \(x\in X(R)\), then \(|o_l(x)|=(|F|^n-1)(|F|^n-|F|)\cdots(|F|^n-|F|^{s-1})\); (Theorem 2.5) if \(R\) is a semisimple Artinian ring with \(X(R)\neq\emptyset\), then \(|o_l(x)|=|o_l(y)|\) for all \(x,y\in X(R)\) if and only if \(R=\bigoplus_{i=1}^m M_{n_i}(D_i)\) with \(D_i\) infinite division rings of the same cardinalities or \(R=M_2(F)\) with a finite field \(F\).NEWLINENEWLINE In Section 3, the authors continue [\textit{Y. Hirano}, Lect. Notes Pure Appl. Math. 236, 343-347 (2004; Zbl 1063.16035)]. They prove that (Theorem 3.3) if a ring \(R\) has exactly two orbits under the left regular action of \(G(R)\) on \(X(R)\), then \(R\) is a local ring or \(R\) is a direct product of two division rings; (Theorem 3.4) a semisimple ring \(R\) has exactly three orbits under the left regular action of \(G(R)\) on \(X(R)\) if and only if \(R=M_2(\mathbb Z_2)\); (Theorem 3.7) if \(R\) is a left Artinian ring with \(G(R)\) cyclic group, then \(R\) is finite. In particular, Theorem 3.7 holds for every ring \(R\) with a finite number of orbits under the left regular action of \(G(R)\) on \(X(R)\).NEWLINENEWLINE Some results in this area are recently obtained by \textit{A. Mȩcel} and \textit{J. Okniński} [Publ. Mat. Barc. 57, No. 2, 477-496 (2013; Zbl 1292.16013)] and \textit{M. Hryniewicka} and \textit{J. Krempa} [Publ. Mat. Barc. 58, No. 1, 233-249 (2014; Zbl 1297.16034)].