Multiplication Function on Ring with Unity is Zero if 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 $g_a: \Z \to R$ 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.

Then:
 * $p \divides n \implies n \cdot a = 0_R$

where $p \divides n$ denotes that $p$ is a divisor of $n$.

Proof
Let $p > 0$.

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


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

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

We have:

Then we have: