Multiple Function on Ring is Zero iff Characteristic is Divisor

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

Then:
 * $n \cdot a = 0_R$


 * $p \divides n$
 * $p \divides n$

where $\cdot$ denotes the multiple operation.

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

Then from Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic:
 * $\map \ker {g_a} = \ideal p$

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

We have by definition of kernel:


 * $n \in \map \ker {g_a} \iff n \cdot a = 0_R$

and by definition of principal ideal:


 * $n \in \ideal p \iff p \divides n$

The result follows.