Ring Epimorphism from Integers to Integers Modulo m

Theorem
Let $\left({\Z, +, \times}\right)$ be the ring of integers.

Let $\left({\Z_m, +_m, \times_m}\right)$ be the ring of integers modulo m.

Let $\phi: \left({\Z, +, \times}\right) \to \left({\Z_m, +_m, \times_m}\right)$ be the mapping defined as:
 * $\forall x \in \Z: \phi \left({x}\right) = \left[\!\left[{x}\right]\!\right]_m$

where $\left[\!\left[{x}\right]\!\right]_m$ is the residue class modulo $m$.

Then $\phi$ is a ring homomorphism.

The image of $\phi$ is $\left({\Z_m, +_m, \times_m}\right)$.

The kernel of $\phi$ is $m \Z$, the set of integer multiples of $m$.