principal ideal domain

主理想环

Algebra & Algebraic Geometry · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: PID

Translation evidence: Literature usage

Meaning in context

A domain whose every ideal is generated by a single element.

每个理想都由单个元素生成的整环。

Example

$F[x]$ is a PID when $F$ is a field.

Usage and distinctions

A ring with principal ideals but zero divisors is not a principal ideal domain.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Wen-Wei Li — Methods in Algebra I, author source交换环初探 (sec:comm-ring-intro), definition def:PID, 主理想环 clause requiring an integral domain · Checked 2026-10-15

Translation sources

1. 李文威 — 代数学方法 卷一，作者源文件交换环初探 (sec:comm-ring-intro), definition def:PID, 主理想环 clause requiring an integral domain · Checked 2026-10-15

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.