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For a skew polynomial ring $R=A[X;\theta,\delta]$ where $A$ is a commutativefrobenius 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 finitecommutative 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 therestriction of $\delta$ and $\theta$ to $B$ are polynomial maps. If a Gr\"obnerbasis can be computed over $B$, then we show that all Euclidean and Hermitiandual-containing codes $\mathcal{C}=Rg/Rf\subset R/Rf$ can be computed using aGr\"obner basis. We also give an algorithm to test if the dual code is again acyclic left module code. We illustrate our approach for rings of order $4$ withnon-trivial endomorphism and the Galois ring of characteristic $4$.

A Gr\"obner Approach to Dual-Containing Cyclic Left Module $(\theta,\delta)$-Codes $Rg/Rf\subset R/Rf$ over Finite Commutative Frobenius Rings