Homomorphism from Integers into Ring with Unity

Theorem
Let $\left({R, +, \circ}\right)$ 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:
 * $\forall n \in \Z: g_a \left({n}\right) = n \cdot a$

Then $g_a$ is a group homomorphism from $\left({\Z, +}\right)$ to $\left({R, +}\right)$.

Also:
 * $\left({p}\right) \subseteq \ker \left({g_a}\right)$

where:
 * $\ker \left({g_a}\right)$ is the kernel of $g_a$;
 * $\left({p}\right)$ is the principal ideal of $\Z$ generated by $p$.

Also:
 * $p \mathop \backslash n \implies n \cdot a = 0$

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

Proof
The fact that $g_a$ is a group homomorphism follows directly from Index Laws for Monoids.

By General Distributivity Theorem, we have that:
 * $\forall n \in \Z_{>0}: \left({n \cdot x} \right) \circ y = n \cdot \left({x \circ y}\right) = x \circ \left({n \cdot y}\right)$

So:
 * $\forall n \in \Z_{>0}: n \cdot a = \left({n \cdot a} \right) \circ 1_R = a \circ \left({n \cdot 1_R}\right)$

So when $n \cdot 1 = 0$ we have $n \cdot a = 0$.