Quotient ring - Imported Knowlpedia Content

Reflexive: a ∼ a a\sim a a ∼ a .
Symmetric: if a ∼ b a\sim b a ∼ b , then b ∼ a b\sim a b ∼ a .
Transitive: if a ∼ b a\sim b a ∼ b and b ∼ c b\sim c b ∼ c , then a ∼ c a\sim c a ∼ c .

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 $R/I\cong R$ .