Definition:Quotient Ring

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Definition

Let $\struct {R, +, \circ}$ be a ring.

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


Let $R / J$ be the coset space of $R$ with respect to $J$.


Let the operation $+$ be defined on $R / J$ by addition of cosets of $J$:

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

Let us also define the operation $\circ$ on $R / J$ by by product of cosets of $J$:

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


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


Examples

Quotient Ring of Integers by Principal Ideal Generated by 3

Let $\struct {\Z, + \times}$ denote the ring of integers.

Let $\struct {3 \Z, +, \times}$ denote the principal ideal of $\Z$ generated by $3$.

Let $\Z / 3 \Z$ denote the quotient ring of $\Z$ by $3 \Z$.


Then $\Z / 3 \Z$ has $3$ elements:

\(\ds 0 + 3 \Z = 3 \Z\) \(=\) \(\ds \set {0, \pm 3, \pm 6, \ldots}\)
\(\ds 1 + 3 \Z\) \(=\) \(\ds \set {\ldots, -5, -2, 1, 4, 7, \ldots}\)
\(\ds 2 + 3 \Z\) \(=\) \(\ds \set {\ldots, -4, -1, 2, 5, 8, \ldots}\)


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

A quotient ring 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.


Sources

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