Noetherian commutative ring

Noether 环

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

Translation evidence: Literature usage

Meaning in context

A commutative ring in which every ascending chain of ideals stabilizes, equivalently every ideal is finitely generated.

每条理想升链最终稳定、等价地每个理想均有限生成的交换环。

Example

Z[x1,…,xn]\mathbb Z[x_1,\ldots,x_n] is Noetherian for finite nn.

Usage and distinctions

A Noetherian ring may have infinitely many ideals; stabilization is not finiteness of their collection.

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Definition sources

  1. Wen-Wei Li — Methods in Algebra I, author source链条件和模的合成列, definition def:ACC-DCC-mod and following ring specialization; lemma prop:fg-noetherian for the equivalent finite-generation condition · Checked 2026-10-15

Translation sources

  1. 李文威 — 代数学方法 卷一，作者源文件链条件和模的合成列, def:ACC-DCC-mod and following paragraph naming Noether 环 · Checked 2026-10-14
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