Ring Direct Product of Modulo Integers is Isomorphic to Ring Modulo Product iff Coprime

Theorem

Let $m, n \in \Z_{>1}$.

Let $\struct {\Z_m, +_m, \times_m}$ and $\struct {\Z_n, +_n, \times_n}$ be the rings of integers modulo $m$ and $n$ respectively.

Let $\struct {\Z_m \times \Z_n}$ be the direct product of $\Z_m$ and $\Z_n$.

Let $\struct {\Z_{m n}, +_{m n}, \times_{m n} }$ be the ring of integers modulo $mn$.

Then $\struct {\Z_m \times \Z_n}$ is isomorphic to $\struct {\Z_{m n}, +_{m n}, \times_{m n} }$ if and only if $m$ and $n$ are coprime.

Proof

This theorem requires a proof. In particular: First a homomorphism is established between $\struct {\Z_m \times \Z_n}$ and $\struct {\Z_m \times \Z_n}$. Then it is demonstrated that $\phi: \struct {\Z_m \times \Z_n} \to \Z_{m n}$ is a bijection. Some of this has already been done, I'm fairly sure. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences