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 $B$-algebras 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，$\theta$为$A$的自同态，$\delta$为$A$的$\theta$-导子，我们考虑循环左模码$\mathcal{C}=Rg/Rf\subset R/Rf$，此处$g$是$R$中$f$的左右公因子。本文利用斜多项式环框架，在$A$为有限交换Frobenius环时推导出其校验矩阵。我们研究自由$B$-代数$A=B[a_1,\ldots,a_s]$，其中$\delta$和$\theta$在$B$上的限制为多项式映射。若能在$B$上计算Gröbner基，则我们证明所有包含欧几里得对偶和埃尔米特对偶的码$\mathcal{C}=Rg/Rf\subset R/Rf$均可通过Gröbner基计算得到。同时给出算法检验对偶码是否仍为循环左模码。我们以4阶环（含非平凡自同态）和特征为4的伽罗瓦环为例说明该方法。