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

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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:

\(\ds \eqclass a m \circ x\) \(=\) \(\ds a \cdot x\) Definition of $\circ$
\(\ds \) \(=\) \(\ds \paren {b + k m} \cdot x\)
\(\ds \) \(=\) \(\ds b \cdot x + k m \cdot x\) Powers of Group Elements: Sum of Indices
\(\ds \) \(=\) \(\ds b \cdot x + k \cdot \paren {m \cdot x}\) Powers of Group Elements: Product of Indices
\(\ds \) \(=\) \(\ds b \cdot x + k \cdot 0_R\) Characteristic times Ring Element is Ring Zero
\(\ds \) \(=\) \(\ds b \cdot x + 0_R\) Power of Identity is Identity
\(\ds \) \(=\) \(\ds b \cdot x\)
\(\ds \) \(=\) \(\ds \eqclass b m \circ x\) Definition of $\circ$

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

$\Box$


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

Module Axiom $\text M 1$: Distributivity over Module Addition

We need to show that:

$\eqclass a m \circ \paren {x + y} = \eqclass a m \circ x + \eqclass a m \circ y$


\(\ds \eqclass a m \circ \paren {x + y}\) \(=\) \(\ds a \cdot \paren {x + y}\) Definition of $\circ$
\(\ds \) \(=\) \(\ds a \cdot x + a \cdot y\) Power of Product in Abelian Group
\(\ds \) \(=\) \(\ds \eqclass a m \circ x + \eqclass a m \circ y\) Definition of $\circ$

$\Box$


Module Axiom $\text M 2$: Distributivity over Scalar Addition

We need to show that:

$\paren {\eqclass a m +_m \eqclass b m} \circ x = \eqclass a m \circ x + \eqclass b m \circ x$


\(\ds \paren {\eqclass a m +_m \eqclass b m} \circ x\) \(=\) \(\ds \eqclass {a + b} m \circ x\) Definition of Modulo Addition
\(\ds \) \(=\) \(\ds \paren {x + y} \cdot x\) Definition of $\circ$
\(\ds \) \(=\) \(\ds a \cdot x + b \cdot x\) Powers of Group Elements: Sum of Indices
\(\ds \) \(=\) \(\ds \eqclass a m \circ x + \eqclass b m \circ x\) Definition of $\circ$

$\Box$


Module Axiom $\text M 3$: Associativity

We need to show that:

$\paren {\eqclass a m \times_m \eqclass b m} \circ x = \eqclass a m \circ \paren {\eqclass b m \circ x}$
\(\ds \paren {\eqclass a m \times_m \eqclass b m} \circ x\) \(=\) \(\ds \eqclass {a \times b} m \circ x\) Definition of Modulo Multiplication
\(\ds \) \(=\) \(\ds \paren {a \times b} \cdot x\) Definition of $\circ$
\(\ds \) \(=\) \(\ds a \cdot \paren {b \cdot x}\) Powers of Group Elements: Product of Indices
\(\ds \) \(=\) \(\ds \eqclass a m \circ \paren {\eqclass b m \circ x}\) Definition of $\circ$

$\Box$


Unitary Module Axiom $\text {UM} 4$: Unity of Scalar Ring

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.

$\Box$


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

$\blacksquare$

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