# Dedekind-Hasse norm implies principal ideal ring

## Contents

## Statement

A commutative unital ring that admits a Dedekind-Hasse norm must be a principal ideal ring.

In particular, an integral domain that admits a Dedekind-Hasse norm must be a principal ideal domain.

## Definitions used

### Dedekind-Hasse norm

`Further information: Dedekind-Hasse norm`

A Dedekind-Hasse norm on a commutative unital ring is a function from the nonzero elements of to the nonnegative integers with the property that for any elements with , it is true that either or there exists an element in the ideal such that .

### Principal ideal ring

`Further information: Principal ideal ring`

A commutative unital ring is termed a principal ideal ring if every ideal of is principal.

## Proof

**Given**: A commutative unital ring with a Dedekind-Hasse norm . An ideal of .

**To prove**: There exists such that .

**Proof**: If , we can take , and we are done.

So, suppose . Consider the function on the nonzero elements of . Since maps to a well-ordered set, there is an element such that for all .

Pick any . If , then or there exists with .

Note that in the latter case, we have , so , and , contradicting the assumption that has minimum norm among the nonzero elements of . Hence, we have the case . Thus, . Conversely, we clearly have , so .