Lattice of Ideals of the Polynomial Ring over a Commutative Chain Ring

Let $R$ be a commutative chain ring. We use a variation of Gr"obner bases to study the lattice of ideals of $R[x]$. Let $I$ be a proper ideal of $R[x]$. We are interested in the following two questions: When is $R[x]\mathrm{/}I$ Frobenius? When is $R[x]\mathrm{/}I$ Frobenius and local? We develop algorithms for answering both questions. When the nilpotency of $\text{rad}R$ is small, the algorithms provide explicit answers to the questions.