What is a principal ideal domain?

A principal ideal domain, or PID, is an integral domain where every ideal is formed by the multiples of a single element. These algebraic structures share many divisibility properties with the integers, including the guarantee that every element has a unique factorization into prime elements. All Euclidean domains and fields are principal ideal domains, and they sit within a specific hierarchy of ring-theoretic classes.

What we know

• A principal ideal domain is an integral domain in which every ideal is principal. [2]
• Principal ideal domains appear in a chain of class inclusions after unique factorization domains. [2, 4]
• Principal ideal domains are Noetherian, integrally closed, unique factorization domains, and Dedekind domains. [2]
• All Euclidean domains and all fields are principal ideal domains. [2]

Still open

• Some authors additionally require PIDs to have Krull dimension one to exclude fields, while most other sources regard fields as trivial special cases.

Where to go next

• the bigger pictureWhat is an integral domain?
• compared withHow do Euclidean domains relate to PIDs?
• a case in pointWhat is a unique factorization domain?
• how it workedWhat is a principal ideal in ring theory?
• what followedWhat are Dedekind domains?
• an unexpected connectionHow do fields fit into principal ideal domains?Although fields are often thought of separately in algebra, they are considered trivial special cases of principal ideal domains by most sources.

Sources

• [1]principal ideal domain (Wikidata Q1143969) · CC0 1.0
• [2]Principal ideal domain · CC BY-SA 4.0
• [3]Principal ideal · CC BY-SA 4.0
• [4]Unique factorization domain · CC BY-SA 4.0
• [5]Integral domain · CC BY-SA 4.0

Your path

1. What was ancient Greek black-figure pottery?
2. /What is a principal ideal domain?