# Definition:Quotient Epimorphism/Ring

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## Definition

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$.

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

Let $\struct {R / J, +, \circ}$ be the quotient ring defined by $J$.

The mapping $\phi: R \to R / J$ given by:

- $\forall x \in R: \map \phi x = x + J$

is known as the **quotient (ring) epimorphism from $\struct {R, +, \circ}$ (on)to $\struct {R / J, +, \circ}$**.

## Also known as

The **quotient (ring) epimorphism** is also known as:

- the
**quotient (ring) morphism** - the
**natural (ring) epimorphism** - the
**natural (ring) morphism** - the
**natural (ring) homomorphism** - the
**canonical (ring) epimorphism** - the
**canonical (ring) morphism**.

In all of the above, the specifier **ring** is usually not used unless it is necessary to distinguish it from a quotient group epimorphism.

## Sources

- 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 60$. Factor rings