Z/(m)-Module Associated with Ring of Characteristic m

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

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

Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$.

Let $\circ$ be the mapping from $\Z_m \times R$ to $R$ defined as:


 * $\forall \eqclass a m \in \Z_m: \forall x \in R: \eqclass a m \circ x = a \cdot x$

where $\eqclass a m$ is the residue class of $a$ modulo $m$ and $a \cdot x$ is the $a$th power of $x$.

Then $\struct {R, +, \circ}_{\Z_m}$ is a unitary $\Z_m$-module.

Proof
Let us verify that the definition of $\circ$ is well-defined.

Let $\eqclass a m = \eqclass b m$.

Then we need to show that:


 * $\forall x \in R: \eqclass a m \circ x = \eqclass b m \circ x$

By the definition of congruence:


 * $\eqclass a m = \eqclass b m \iff \exists k \in \Z : a = b + k m$

Then:

Thus, the definition of $\circ$ is well-defined.

Let us verify that $\struct {R, +, \circ}_{\Z_m}$ is a unitary $\Z_m$-module by verifying the axioms in turn.

We need to show that:
 * $\eqclass a m \circ \paren {x + y} = \eqclass a m \circ x + \eqclass a m \circ y$

We need to show that:
 * $\paren {\eqclass a m +_m \eqclass b m} \circ x = \eqclass a m \circ x + \eqclass b m \circ x$

We need to show that:
 * $\paren {\eqclass a m \times_m \eqclass b m} \circ x = \eqclass a m \circ \paren {\eqclass b m \circ x}$

We need to show that:
 * $\eqclass 1 m \circ x = x$

since $\eqclass 1 m$ is the unity of $\Z_m$.

That is, that $1 \cdot x = x$.

This follows from the definition of power of group element.

Having verified all four axioms, we have shown that $\struct {R, +, \circ}_{\Z_m}$ is a unitary $\Z_m$-module.