prime ideal

素理想

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

Translation evidence: Literature usage

Meaning in context

A proper ideal $\mathfrak p$ in a commutative ring with $ab\in\mathfrak p$ implying $a\in\mathfrak p$ or $b\in\mathfrak p$.

交换环中包含乘积就必包含某一因子的真理想。

Example

$(0)$ is prime in $\mathbb Z$.

Usage and distinctions

Properness matters; the whole ring is never a prime ideal.

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:ring-ideal, first item 素理想, proper-ideal hypothesis and product condition · 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, first item 素理想, proper-ideal hypothesis and product condition · Checked 2026-10-15

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