Kernel is Trivial iff Monomorphism/Ring

Theorem

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

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

Then $\phi$ is a ring monomorphism if and only if $\map \ker \phi = 0_{R_1}$.

Proof

The proof for the ring monomorphism follows directly from:

Ring Homomorphism of Addition is Group Homomorphism

and:

Kernel is Trivial iff Group Monomorphism for the group monomorphism.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 24$. Homomorphisms: Theorem $45$
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms: Lemma $2.6 \ \text{(i)}$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 57.2$ Ring homomorphisms