quotient ring

商环

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

Translation evidence: Literature usage

Meaning in context

For an ideal II of a commutative ring RR, the cosets R/IR/I with addition and multiplication induced by representatives.

对交换环理想 II，在陪集 R/IR/I 上由代表元运算诱导得到的环。

Example

R[x]/(x2+1)≅C\mathbb R[x]/(x^2+1)\cong\mathbb C.

Usage and distinctions

For noncommutative rings a quotient ring requires a two-sided ideal.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Wen-Wei Li — Methods in Algebra I, author source基本概念 (sec:ring-basics), definition naming 商环 after the ideal operations, (r+I)(s+I)=rs+I and induced additive quotient · Checked 2026-10-15

Translation sources

  1. 李文威 — 代数学方法 卷一，作者源文件基本概念 (sec:ring-basics), definition naming 商环 after the ideal operations, (r+I)(s+I)=rs+I and induced additive quotient · Checked 2026-10-15
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