Quotient Ring of Kernel of Ring Epimorphism

Theorem

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring epimorphism.

Let $K = \map \ker \phi$ be the kernel of $\phi$.


There exists a unique ring isomorphism $g: R_1 / K \to R_2$ such that:

$g \circ q_K = \phi$


Proof

From the Quotient Theorem for Epimorphisms, there is one and only one isomorphism that satisfies the conditions for each of the operations on $R_1$.

Hence the result.

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.6: \ 1^\circ$
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