Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic

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 $p$.

Let $g_a: \Z \to R$ be the mapping from the integers into $R$ defined as:
 * $\forall n \in \Z:\forall a \in R: \map {g_a} n = n \cdot a$

where $\cdot$ denotes the multiple operation.

Let $a \in R$ such that $a$ is not a zero divisor of $R$.

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

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

Proof
From Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function we have:
 * $\ideal p \subseteq \map \ker {g_a}$

for all $a \in R$.

It remains to be shown that for all $a \in R$ such that $a$ is not a zero divisor of $R$:
 * $\map \ker {g_a} \subseteq \ideal p$

So:

Hence the result by definition of subset.