Convolutional codes with a matrix-algebra word-ambient

Let $\mathcal{M}_n(\mathbb{F})$be the algebra of \(n \times n\) matrices over the finite field $\mathbb{F}$. In this paper we prove that the dual code of each ideal convolutional code in the skew-polynomial ring $\mathcal{M}_n(\mathbb{F})[z;\sigma_U]$ which is a direct summand as a left ideal, is also an ideal convolutional code over $\mathcal{M}_n(\mathbb{F})[z;\sigma_UT]$ and a direct summand as a left ideal. Moreover we provide an algorithm to decide if \(\sigma_U\) is a separable automorphism and returns the corresponding separability element, when pertinent.