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