# Quotient Group of Ideal is Coset Space

## Theorem

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

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

Let $\struct {R / J, +}$ be the quotient group of $\struct {R, +}$ by $\struct {J, +}$.

Then each element of $\struct {R / J, +}$ is a coset of $J$ in $R$, that is, is of the form $x + J = \set {x + j: j \in J}$ for some $x \in R$.

The rule of addition of these cosets is: $\paren {x + J} + \paren {y + J} = \paren {x + y} + J$.

The identity of $\struct {R / J, +}$ is $J$ and for each $x \in R$, the inverse of $x + J$ is $\paren {-x} + J$.

## Proof

From Ideal is Additive Normal Subgroup that $J$ is a normal subgroup of $R$ under $+$.

From Quotient Ring Addition is Well-Defined, $+$ is a well-defined operation.

The rest follows directly from the definition of quotient group.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 60$. Factor rings