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

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

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

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

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


 * $\forall \left[\!\left[a\right]\!\right]_m \in \Z_m: \forall x \in R: \left[\!\left[a\right]\!\right]_m \circ x = a \cdot x$

where $\left[\!\left[a\right]\!\right]_m$ is the residue class of $a$ modulo $m$ and $a \cdot x$ is the $a$th power of $x$.

Then $\left({R, +, \circ}\right)_{\Z_m}$ is a unitary $\Z_m$-module.

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

Let $\left[\!\left[a\right]\!\right]_m=\left[\!\left[b\right]\!\right]_m$.

Then we need to show that:


 * $\forall x \in R : \left[\!\left[a\right]\!\right]_m \circ x = \left[\!\left[b\right]\!\right]_m \circ x$

By the definition of congruence:


 * $\left[\!\left[a\right]\!\right]_m=\left[\!\left[b\right]\!\right]_m \iff \exists k \in \Z : a=b + km$

Then:

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

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

Axiom $(1)$
We need to show that $\left[\!\left[a\right]\!\right]_m \circ \left({x + y}\right) = {\left[\!\left[a\right]\!\right]_m \circ x} + {\left[\!\left[a\right]\!\right]_m \circ y}$.

Axiom $(2)$
We need to show that $\left( {\left[\!\left[a\right]\!\right]_m +_m \left[\!\left[b\right]\!\right]_m}\right) \circ x = \left[\!\left[a\right]\!\right]_m \circ x + \left[\!\left[b\right]\!\right]_m \circ x$.

Axiom $(3)$
We need to show that $\left({\left[\!\left[a\right]\!\right]_m \times_m \left[\!\left[b\right]\!\right]_m}\right) \circ x = \left[\!\left[a\right]\!\right]_m \circ \left({\left[\!\left[b\right]\!\right]_m \circ x}\right)$.

Axiom $(4)$
We need to show that $\left[\!\left[1\right]\!\right]_m \circ x = x$, since $\left[\!\left[1\right]\!\right]_m$ 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 $\left({R, +, \circ}\right)_{\Z_m}$ is a unitary $\Z_m$-module.