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

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