quotient ring

《代数学引论》 · 第 51 页 · 定义 16

设 $I$ 是环 $L$ 的一个理想。$L$ 对于 $I$ 的陪集在上面定义的运算下所成的环称为 $L$ 对于 $I$ 的商环，记为 $L/I$。

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《Abstract Algebra》 · 第 242—243 页 · Section 7.3「Ring Homomorphisms and Quotient Rings」，Proposition 6 及 Definition

Proposition 6. Let $R$ be a ring and let $I$ be an ideal of $R$. Then the (additive) quotient group $R/I$ is a ring under the binary operations: $(r+I)+(s+I)=(r+s)+I$ and $(r+I)\times(s+I)=(rs)+I$ for all $r,s\in R$. Conversely, if $I$ is any subgroup such that the above operations are well defined, then $I$ is an ideal of $R$. Definition. When $I$ is an ideal of $R$ the ring $R/I$ with the operations in the previous proposition is called the quotient ring of $R$ by $I$.