Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function

Theorem
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let the characteristic of $R$ be $p$.

For any $a \in R$, we define the mapping $g_a: \Z \to R$ from the integers into $R$ as:
 * $\forall n \in \Z: \map {g_a} n = n \cdot a$

Then:
 * $\ideal p \subseteq \map \ker {g_a}$

where:
 * $\map \ker {g_a}$ is the kernel of $g_a$
 * $\ideal p$ is the principal ideal of $\Z$ generated by $p$.

Proof
We have from Multiple Function on Ring is Homomorphism that $g_a$ is a group homomorphism.

By definition of kernel:
 * $x \in \map \ker {g_a} \iff \map {g_a} x = 0_R$

Hence to show that $\ideal p \subseteq \map \ker {g_a}$, we need to show that:
 * $\forall x \in \ideal p: \map {g_a} x = 0_R$

By definition of characteristic, $p \in \Z_{\ge 0}$ is such that $\ideal p$ is the kernel of $g_1$:
 * $\map {g_1} n = n \cdot 1_R$

So:

Then:

So:
 * $x \in \map \ker {g_a}$

and so:
 * $\ideal p \subseteq \map \ker {g_a}$