Composition of Ring Monomorphisms is Ring Monomorphism

Theorem

Let:

$\struct {R_1, +_1, \circ_1}$

$\struct {R_2, +_2, \circ_2}$

$\struct {R_3, +_3, \circ_3}$

be rings.

Let:

$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$

$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$

be (ring) monomorphisms.

Then the composite of $\phi$ and $\psi$ is also a (ring) monomorphism.

Proof

A ring monomorphism is a ring homomorphism which is also an injection.

From Composition of Ring Homomorphisms is Ring Homomorphism, $\psi \circ \phi$ is a ring homomorphism.

From Composite of Injections is Injection, $\psi \circ \phi$ is an injection.

$\blacksquare$

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms: Definition $2.4$