Homomorphism from Integers into Ring with Unity

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 the following hold: