Definition:Quotient Ring

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

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

Let $R / J$ be the (right) 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 $J$.

At this point, it is not even ensured that $+$ and $\circ$ are well-defined, but this can be shown.

In fact, $\left({R / J, +, \circ}\right)$ is a ring, as shown on Quotient Ring is Ring.

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

Also see
In Quotient Ring is an Ideal it is proved not only that $\left({R / J, +, \circ}\right)$ is a ring, but also that it is an ideal of $R$.