# Polynomial Rings Over Commutative Reduced Hopfian Local Rings

## TL;DR

This paper characterizes when polynomial rings over commutative, reduced, local rings are Hopfian, showing the equivalence with the base ring, and provides examples of non-Noetherian Hopfian domains.

## Contribution

It proves that for reduced local rings, the Hopfian property is preserved under polynomial extension, answering a specific open question and expanding understanding of Hopfian rings.

## Key findings

- R is Hopfian iff R[x] is Hopfian for reduced local rings
- Finite dimensional domains are Hopfian
- Provides examples of non-Noetherian Hopfian domains

## Abstract

In this paper we prove that if $R$ is a commutative, reduced, local ring, then $R$ is Hopfian if and only if the ring $R[x]$ is Hopfian. This answers a question of Varadarajan, in the case when $R$ is a reduced local ring. We provide examples of non-Noetherian Hopfian commutative domains by proving that the finite dimensional domains are Hopfian. Also, we derive some general results related to Hopfian rings.