# PID implies one-dimensional

From Commalg

This article gives the statement and possibly, proof, of an implication relation between two integral domain properties. That is, it states that every integral domain satisfying the first integral domain property (i.e., principal ideal domain) must also satisfy the second integral domain property (i.e., one-dimensional domain)

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## Contents

## Statement

A principal ideal domain is a one-dimensional domain. In other words, in a principal ideal domain, every nonzero prime ideal is a maximal ideal.

## Related facts

- Principal ideal ring implies one-dimensional
- Unique factorization and one-dimensional iff principal ideal
- Unique factorization implies every nonzero prime ideal contains a prime element
- Unique factorization and finite-dimensional implies every prime ideal is generated by a set of primes of size at most the dimension

## Facts used

## Proof

**Given**: A principal ideal domain , a nonzero prime ideal of .

**To prove**: is a maximal ideal of .

**Proof**: Suppose is not maximal in . Then, by fact (1), there exists a maximal ideal of properly containing . Since is a principal ideal domain, there exist such that . In particular, we can write for some .

Note that since is prime, and because is properly contained in , we have . Thus, for some . This gives , giving . Since is an integral domain, this forces either (a contradiction to being nonzero) or (a contradiction to being a proper ideal). Thus, we have the required contradiction.