# Definition:Quotient Epimorphism/Ring

## 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.