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

TL;DR

This paper investigates the structure of ideals in polynomial rings over commutative chain rings, focusing on conditions for the quotient to be Frobenius and local, using modified Gr"obner bases and algorithms.

Contribution

It introduces algorithms based on a variation of Gr"obner bases to determine when polynomial ring quotients over chain rings are Frobenius and local.

Findings

Algorithms for identifying Frobenius quotients

Explicit criteria when the nilpotency is small

Enhanced understanding of ideal lattice structure

Abstract

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.