Modulo Multiplication is Associative

Theorem

Multiplication modulo $m$ is associative:

$\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m, \left[\!\left[{z}\right]\!\right]_m \in \Z_m: \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m}\right) \times_m \left[\!\left[{z}\right]\!\right]_m = \left[\!\left[{x}\right]\!\right]_m \times_m \left({\left[\!\left[{y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)$.

Proof

Follows directly from the definition of multiplication modulo $m$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m}\right) \times_m \left[\!\left[{z}\right]\!\right]_m\)	\(=\)	\(\displaystyle \left[\!\left[{x y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left({x y}\right) z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x \left({y z}\right)}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x}\right]\!\right]_m \times_m \left({\left[\!\left[{y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 2$: Example $2.3$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 19.1$