maximal ideal

极大理想

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

Translation evidence: Literature usage

Meaning in context

A proper ideal not strictly contained in any other proper ideal of the ring.

不严格包含于任何其他真理想的真理想。

Example

(5)(5) is maximal in Z\mathbb Z.

Usage and distinctions

Every maximal ideal of a commutative ring is prime, but (0)(0) in Z\mathbb Z is not maximal.

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), proof of prop:existence-maximal-ideal, paragraph identifying maximal elements among proper ideals with 极大理想 · 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:ring-ideal, second item 极大理想; proper-ideal clarification in proof of prop:existence-maximal-ideal · Checked 2026-10-15
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