Definition:Quotient Ring

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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 $J$.

This is also known as a factor ring.