On the multiplicative monoid of \(n\times n\) matrices over \(\mathbb{Z}_{p^m}\). (Q2463896)

For any positive integers \(m\), \(n\) and a prime number \(p\), the author introduces a concept of invariant factors for matrices over \(\mathbb{Z}_{p^m}\) first, then investigates the canonical Green's relations in \(M_n(\mathbb{Z}_{p^m})\), the Schützenberger groups of \(\mathcal D\)-classes, group \(\mathcal H\)-classes and regular principal factors of the full matrix monoid \(M_n(\mathbb{Z}_{p^m})\). These studies result in a characterization of a matrix which has a group inverse in \(M_n(\mathbb{Z}_{p^m})\). Observe that any finite monoid can be regarded as a linear algebraic monoid, in particular, \(M_n(\mathbb{Z}_{p^m})\) is a ``standard'' finite monoid of Lie type, even a finite reductive monoid. The reviewer would like to point out that some of the results about the Green's relations in \(M_n(\mathbb{Z}_{p^m})\) can be obtained by the Putcha-Renner theories of reductive monoids and linear algebraic monoids.