Quotient Epimorphism is Epimorphism

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Theorem

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 $J$.