Ring Epimorphism with Trivial Kernel is Isomorphism

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 $\phi$.

$\phi$ is an isomorphism if and only if $K = \set {0_{R_1} }$.

Proof

From Kernel is Trivial iff Monomorphism, $\phi$ is a ring monomorphism if and only if $K = \set {0_{R_1} }$.

As $\phi$ is also an epimorphism, the result follows.

$\blacksquare$

Sources

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