# 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 (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

- Quotient Ring Addition is Well-Defined
- Quotient Ring Product is Well-Defined
- Quotient Ring is Ring
- Congruence Relation and Ideal are Equivalent

- Results about
**Quotient Rings**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 22$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Theorem $42$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 60$. Factor rings - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**quotient ring** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**quotient ring**