Exercise 1579Number theory › CongruencesL1Added 3 Nov 2026

The CRT as a ring isomorphism

#CRT#ring isomorphism#product

Statement

Show that the CRT gives a ring isomorphism $\mathbb{Z}/mn\mathbb{Z}\cong\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}$ when $\gcd(m,n)=1$ .

Hint

Define $\varphi:x\bmod{mn}\mapsto(x\bmod m,x\bmod n)$ .

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Solution

Homomorphism. $\varphi$ preserves $+$ and $\times$ component-wise, with $\varphi(1)=(\bar1,\bar1)$ .

Injective. If $\varphi(\bar x)=(0,0)$ , then $m\mid x$ and $n\mid x$ , and $\gcd(m,n)=1$ implies $mn\mid x$ .

Surjective. Both sides have $mn$ elements, and injectivity gives surjectivity.

Conclusion. $\varphi$ is a ring isomorphism.

Application. $\mathbb{Z}/6\mathbb{Z}\cong\mathbb{F}_2\times\mathbb{F}_3$ ; $\mathbb{Z}/8\mathbb{Z}\not\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ (since $\gcd(2,4)=2\neq1$ ).

The CRT as a ring isomorphism

Statement

Solution

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