Quotient Ring

algebra

Definition

Quotient Ring

Let $R$ be a ring and let $I\subseteq R$ be an ideal, that is, an additive subgroup of $R$ that is closed under multiplication by elements of $R$. The quotient ring of $R$ modulo $I$ is the set of cosets

$$R/I:=\{a+I\mid a\in R\}.$$

Addition and multiplication are defined by

$$(a+I)+(b+I):=(a+b)+I,$$ $$(a+I)(b+I):=ab+I.$$

These operations are well-defined because $I$ is an ideal.

Example

For the polynomial ring $\mathbb{R}[x]$ and the ideal $(x^2+1)$,

$$\mathbb{R}[x]/(x^2+1)\cong \mathbb{C}.$$

This identifies the residue class of $x$ with the imaginary unit $i$.