Definition:Quotient Epimorphism/Ring

Definition
Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0_R$ and whose unity is $1_R$.

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

Let $\left({R / J, +, \circ}\right)$ be the quotient ring defined by $J$.

The mapping $\phi: R \to R / J$ given by:
 * $x \in R: \phi \left({x}\right) = x + J$

is known as the natural (ring) epimorphism from $\left({R, +, \circ}\right)$ onto $\left({R / J, +, \circ}\right)$.

Also known as
The natural (ring) epimorphism is also known as:
 * the natural (ring) morphism
 * the natural (ring) homomorphism
 * the quotient (ring) epimorphism
 * the quotient (ring) morphism
 * 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 natural group epimorphism.