Definition:Quotient Ring

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $J$ be an ideal of $R$.

Let $R / J$ be the (left) coset space of $R$ modulo $J$ with respect to $+$.

Define an operation $+$ on $R / J$ by:


 * $\forall x,y: \left({x + J}\right) + \left({y + J}\right) := \left({x + y}\right) + J$

Also, define the operation $\circ$ on $R / J$ by:


 * $\forall x,y: \left({x + J}\right) \circ \left({y + J}\right) := \left({x \circ y}\right) + J$

The algebraic structure $\left({R / J, +, \circ}\right)$ is called the quotient ring of $R$ by $J$.

Also denoted as
While the inline form of the fraction notation $R / J$ is usually used for a quotient ring, some presentations use the full $\dfrac R J$ form.

It is usual for the latter form to be used only when either of both of the expressions top and bottom are more complex than single symbols.

Also known as
This is also known as a factor ring.

Some sources refer to this as a residue class ring.

Also see

 * Quotient Ring Addition is Well-Defined
 * Quotient Ring Product is Well-Defined
 * Quotient Ring is Ring
 * Congruence Relation and Ideal are Equivalent