  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 $R[x]/I$ Frobenius? When
is $R[x]/I$ 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.
