Definition:Quotient Ring

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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.

Linguistic Note

The word quotient derives from the Latin word meaning how often.