Category:Quotient Epimorphism is Epimorphism
Jump to navigation
Jump to search
This category contains pages concerning Quotient Epimorphism is Epimorphism:
Group
Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $G / N$ be the quotient group of $G$ by $N$.
Let $q_N: G \to G / N$ be the quotient epimorphism from $G$ to $G / N$:
- $\forall x \in G: \map {q_N} x = x N$
Then $q_N$ is a group epimorphism whose kernel is $N$.
Ring
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 $J$.
Let $\phi: R \to R / J$ be the quotient (ring) epimorphism from $R$ to $R / J$:
- $x \in R: \map \phi x = x + J$
Then $\phi$ is a ring epimorphism whose kernel is $J$.
Pages in category "Quotient Epimorphism is Epimorphism"
The following 3 pages are in this category, out of 3 total.