Kernel of Multiple Function on Ring with Characteristic Zero is Trivial

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

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 $R$.

Let the characteristic of $R$ be $0$.

Then:
 * $\map \ker {g_a} = \set {0_R}$

where $\ker$ denotes the kernel of $g_a$.

That is:
 * $n \cdot a = 0_R$


 * $n = 0$
 * $n = 0$

Proof
For $n = 0$, we trivially have $n \cdot a = 0_R$.

$\exists n \ne 0: n \cdot a = 0_R$.

Then:

This contradicts our assertion that the characteristic of $R$ is $0$.

Hence by Proof by Contradiction there can be no such $n \ne 0$ such that $n \cdot a = 0_R$.