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

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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:

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

Then:

\(\displaystyle \left[\!\left[a\right]\!\right]_m \circ x\) \(=\) \(\displaystyle a \cdot x\) Definition of $\circ$
\(\displaystyle \) \(=\) \(\displaystyle \left(b+km\right) \cdot x\)
\(\displaystyle \) \(=\) \(\displaystyle b \cdot x + km \cdot x\) Powers of Group Elements: Sum of Indices
\(\displaystyle \) \(=\) \(\displaystyle b \cdot x + k \cdot \left( m \cdot x \right)\) Powers of Group Elements: Product of Indices
\(\displaystyle \) \(=\) \(\displaystyle b \cdot x + k \cdot 0_R\) Characteristic times Ring Element is Ring Zero
\(\displaystyle \) \(=\) \(\displaystyle b \cdot x + 0_R\) Power of Identity is Identity
\(\displaystyle \) \(=\) \(\displaystyle b \cdot x\)
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[b\right]\!\right]_m \circ x\) Definition of $\circ$

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

$\Box$


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:

$\left[\!\left[{a}\right]\!\right]_m \circ \left({x + y}\right) = {\left[\!\left[{a}\right]\!\right]_m \circ x} + {\left[\!\left[{a}\right]\!\right]_m \circ y}$


\(\displaystyle \left[\!\left[a\right]\!\right]_m \circ \left({x + y}\right)\) \(=\) \(\displaystyle a \cdot \left(x +y\right)\) Definition of $\circ$
\(\displaystyle \) \(=\) \(\displaystyle a \cdot x + a \cdot y\) Power of Product in Abelian Group
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[a\right]\!\right]_m \circ x + \left[\!\left[a\right]\!\right]_m \circ y\) Definition of $\circ$

$\Box$


Axiom $(2)$

We need to show that:

$\left( {\left[\!\left[a\right]\!\right]_m +_m \left[\!\left[b\right]\!\right]_m}\right) \circ x = \left[\!\left[a\right]\!\right]_m \circ x + \left[\!\left[b\right]\!\right]_m \circ x$


\(\displaystyle \left( {\left[\!\left[a\right]\!\right]_m +_m \left[\!\left[b\right]\!\right]_m}\right) \circ x\) \(=\) \(\displaystyle \left[\!\left[ a+b \right]\!\right]_m \circ x\) Definition of Modulo Addition
\(\displaystyle \) \(=\) \(\displaystyle \left(a+b\right) \cdot x\) Definition of $\circ$
\(\displaystyle \) \(=\) \(\displaystyle a \cdot x + b \cdot x\) Powers of Group Elements: Sum of Indices
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[a\right]\!\right]_m \circ x + \left[\!\left[b\right]\!\right]_m \circ x\) Definition of $\circ$

$\Box$


Axiom $(3)$

We need to show that:

$\left({\left[\!\left[a\right]\!\right]_m \times_m \left[\!\left[b\right]\!\right]_m}\right) \circ x = \left[\!\left[a\right]\!\right]_m \circ \left({\left[\!\left[b\right]\!\right]_m \circ x}\right)$
\(\displaystyle \left({\left[\!\left[a\right]\!\right]_m \times_m \left[\!\left[b\right]\!\right]_m}\right) \circ x\) \(=\) \(\displaystyle \left[\!\left[ a \times b \right]\!\right]_m \circ x\) Definition of Modulo Multiplication
\(\displaystyle \) \(=\) \(\displaystyle \left(a \times b\right) \cdot x\) Definition of $\circ$
\(\displaystyle \) \(=\) \(\displaystyle a \cdot \left({b \cdot x}\right)\) Powers of Group Elements: Product of Indices
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[a\right]\!\right]_m \circ \left({\left[\!\left[b\right]\!\right]_m \circ x}\right)\) Definition of $\circ$

$\Box$


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.

$\Box$


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

$\blacksquare$