# Bounding the length in a module of evaluated skew polynomials

Let $$R$$ be a finite principal ideal ring, $$S$$ a Galois extension of $$R$$ of degree $$m$$ (so in particular $$S$$ is a free $$R$$-module of rank $$m$$, and we have an $$R$$-module isomorphism $$S^n \cong \operatorname{Mat}_{m \times n}(R)$$ (1)). Let $$M$$ be an $$R$$-module and $$\lambda_R(M)$$ its length over $$R$$.

Let $$S[x, \sigma]$$ the ring of skew polynomials over $$S$$ with automorphism $$\sigma$$. Let $$g = (g_1, \dots , g_n) \in S^n$$, where $$\{g_1, \dots , g_n\}$$ is an $$R$$-lin.indep. set of elements of $$S$$, and $$\operatorname{Gab}_k(g)=\{f(g) \ | \ f \in S[x, \sigma], \ \operatorname{deg}(f) , so in particular, by (1), $$\operatorname{Gab}_k(g)$$ can be viewed as an $$R$$-module of $$m \times n$$ matrices. Furthermore, it can be shown that $$\operatorname{Gab}_k(g)$$ has a generator matrix $$\textbf{G}$$ whose $$k$$ rows are $$\sigma^i(g_1) \ \sigma^i(g_2) \ \dots \ \sigma^i(g_n)$$, for $$i=0, \dots, k-1$$.

Is there a way to compute $$\operatorname{min}\{\lambda_R(\operatorname{rowsp}(A)) \ | \ A \in \operatorname{Gab}_k(g)\}$$, where $$\lambda_R(\operatorname{rowsp}(A))$$ denotes the length of the module generated by the rows of the matrix $$A$$? This is closely related to the well-known results about Gabidulin codes over fields, but the techniques used for the latter cannot be applied in this case. I feel that it should not be very difficult, but I don't have much expertise with skew polynomials.