$(\mathbb{Z}/m\mathbb{Z}) \otimes (\mathbb{Z}/n\mathbb{Z}) = 0$ if $m, n$ are coprime
by Hidenori
Proposition
$(\mathbb{Z}/m\mathbb{Z}) \otimes (\mathbb{Z}/n\mathbb{Z}) = 0$ if $m, n$ are coprime.
Solution
If $m, n$ are coprime, there must exist integers $p, q$ such that $pm + qn = 1$.
Let $a \otimes b \in (\mathbb{Z}/m\mathbb{Z}) \otimes (\mathbb{Z}/n\mathbb{Z})$.
\[\begin{align*} a \otimes b &= 1(a \otimes b) \\ &= (pm + qn)(a \otimes b) \\ &= pm(a \otimes b) + qn(a \otimes b) \\ &= p((ma) \otimes b) + q(a \otimes (nb)) \\ &= p(0 \otimes b) + q(a \otimes 0) \\ &= 0 + 0 \\ &= 0. \end{align*}\]Subscribe via RSS