Quotient ring

A ring formed from a ring by identifying elements that differ by a two-sided ideal.

Quotient ring

Let $R$ be a ring and let $I$ be a two-sided ideal of $R$ . The quotient ring $R/I$ is the quotient set of $R$ by the equivalence relation $r\sim s$ iff $r-s\in I$ , with addition and multiplication on cosets defined by

(r+I)+(s+I)=(r+s)+I,\qquad (r+I)(s+I)=(rs)+I.

The two-sided condition ensures multiplication is well-defined.

The canonical projection $\pi:R\to R/I$ is a surjective ring homomorphism with kernel $I, and$ R/I$ satisfies the universal property of quotients .

Examples:

$\mathbb Z/(n)$ is the familiar ring $\mathbb Z/n\mathbb Z$ .
For a field $k$ , the quotient $k[x]/(x^2)$ is a ring in which the class of $x$ is nilpotent.
If $I=R$ , then $R/I$ is the zero ring; if $I=\{0\}$ , then $R/I\cong R$ .