Factor rings of skew polynomial rings over a finite field (Q343241)

Let \(k\) be a finite field of cardinality \(p^r\) and \(\sigma\) an automorphism of \(k\). Certain properties of the factor ring \(S=k[x,\sigma]/(x^n)\) \(n\geq 1\), of the skew polynomial ring \(k[x,\sigma]\) are considered. In particular, it is shown that if \(F\) is a free right \(S\)-module with a basis of \(t\) elements, then any submodule of \(F\) is a direct sum of at most \(t\) cyclic submodules. Moreover, if \(M\) is a finitely generated right \(S\)-module, then \(M\) is a direct sum of cyclic submodules. It is also shown that every local ring \(R\) of characteristic \(p\) and cardinality \(p^{nr}\), with radical \(J\) of cardinality \(p^{r(n-1)}\) and such that \(J^{n-1}\neq 0\), is isomorphic to a ring of the above considered type.