Lattice of ideals of the polynomial ring over a commutative chain ring

Abstract

Let $$R$$ be a commutative chain ring. We use a variation of Gröbner 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.