# Definition:Quotient Ring

## Definition

Let $\struct {R, +, \circ}$ 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:

$\forall x, y: \paren {x + J} + \paren {y + J} := \paren {x + y} + J$

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

$\forall x, y: \paren {x + J} \circ \paren {y + J} := \paren {x \circ y} + J$

The algebraic structure $\struct {R / J, +, \circ}$ 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

• Results about Quotient Rings can be found here.