For a skew polynomial ring $R=A[X;\theta,\delta]$ where $A$ is a commutative Frobenius ring, $\theta$ an endomorphism of $A$ and $\delta$ a $\theta$-derivation of $A$, we consider cyclic left module codes $\mathcal{C}=Rg/Rf\subset R/Rf$ where $g$ is a left and right divisor of $f$ in $R$. In this paper, we derive a parity check matrix when $A$ is a finite commutative Frobenius ring using only the framework of skew polynomial rings. We consider rings $A=B[a_1,\ldots,a_s]$ which are free $B$-modules where the restriction of $\delta$ and $\theta$ to $B$ are polynomial maps. If a Gr\"obner basis can be computed over $B$, then we show that all Euclidean and Hermitian dual-containing codes $\mathcal{C}=Rg/Rf\subset R/Rf$ can be computed using a Gr\"obner basis. We also give an algorithm to test if the dual code is again a cyclic left module code. We illustrate our approach for rings of order $4$ with non-trivial endomorphism and the Galois ring of characteristic $4$.

翻译：对于斜多项式环$R=A[X;\theta,\delta]$，其中$A$为交换Frobenius环，$\theta$是$A$的自同态，$\delta$是$A$的$\theta$-导子，我们考虑循环左模范畴$\mathcal{C}=Rg/Rf\subset R/Rf$，其中$g$是$f$在$R$中的左右因子。本文在$A$为有限交换Frobenius环的情形下，仅利用斜多项式环的框架推导出校验矩阵。我们研究环$A=B[a_1,\ldots,a_s]$（作为自由$B$-模）且$\delta$与$\theta$在$B$上的限制为多项式映射的情形。若能在$B$上计算Gröbner基，则证明所有欧几里得对偶与埃尔米特对偶包含码$\mathcal{C}=Rg/Rf\subset R/Rf$均可通过Gröbner基计算得到。同时给出判定对偶码是否仍为循环左模范畴的算法。我们通过具有非平凡自同态的4阶环及特征为4的Galois环的具体案例演示所提方法。