Modulo Multiplication is Associative/Proof 2
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Theorem
Multiplication modulo $m$ is associative:
- $\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m \times_m \eqclass y m} \times_m \eqclass z m = \eqclass x m \times_m \paren {\eqclass y m \times_m \eqclass z m}$
That is:
- $\forall x, y, z \in \Z_m: \paren {x \cdot_m y} \cdot_m z = x \cdot_m \paren {y \cdot_m z}$
Proof
Let $j$ be the largest integer such that:
- $j m \le x y$
Let $p$ be the largest integer such that:
- $p m \le y z$
By definition of multiplication modulo $m$:
- $x \cdot_m y = x y - j m$
- $y \cdot_m z = y z - p m$
Let $k$ be the largest integer such that:
- $k m \le \paren {x y - j m} z$
Let $q$ be the largest integer such that:
- $q m \le x \paren {y z - p m}$
Then:
- $\paren {j z + k} m \le \paren {x y} z$
- $\paren {q + x p} m \le x \paren {y z}$
Thus:
\(\ds \paren {x \cdot_m y} \cdot_m z\) | \(=\) | \(\ds \paren {x y - j m} z - k m\) | Definition of Modulo Multiplication | |||||||||||
\(\ds x \cdot_m \paren {y \cdot_m z}\) | \(=\) | \(\ds x \left({y z - p m}\right) - q m\) | Definition of Modulo Multiplication |
But suppose that there exists an integer $s$ such that:
- $s m \le \paren {x y} z$
and:
- $j z + k < s$
Then:
- $\paren {j z + k + 1} m \le \paren {x y} z$
from which:
- $\paren {k + 1} m \le \paren {x y - j m} z$
But this contradicts the definition of $k$.
Thus $j z + k$ is the largest of those integers $i$ such that $i m \le \paren {x y} z$.
Similarly, $q + x p$ is the largest of those integers $i$ such that $i m \le x \paren {y z}$.
From Integer Multiplication is Associative:
- $\paren {x y} z = x \paren {y z}$
Thus $j z + k = q + x p$ and so:
\(\ds \paren {x \cdot_m y} \cdot_m z\) | \(=\) | \(\ds \paren {x y - j m} z - k m\) | Definition of Modulo Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds x y z - \paren {j z + k} m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x y z - \paren {q + x p} m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \paren {y z - p m} - q m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \cdot_m \paren {y \cdot_m z}\) | Definition of Modulo Multiplication |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.3$