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Every finite commutative ring $R$ must be of the form $\frac{ \mathbb{Z} }{n_1 \mathbb{Z}} \times \frac{\mathbb{Z}}{n_2 \mathbb{Z}} \times \dots \frac{\mathbb{Z}}{n_k \mathbb{Z}}$.

I am able to proceed upto the step that it should be direct product of artinian commutative local rings. I used structure theorem for commutative ring. But I am unable to proceed further.

GraduateStudent
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    Consider the field with 4 elements. It is not of the form that you desire. – alephnot Oct 11 '20 at 19:04
  • @alephnot Then what does this link https://math.stackexchange.com/questions/305824/structure-of-finite-commutative-rings#:~:text=A%20finite%20commutative%20ring%20is,n%E2%86%A6n1R).&text=It%20follows%20that%20Z%2Fn,holds%20for%20Z%2FnZ say? – GraduateStudent Oct 12 '20 at 04:12
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    @SunShine "finite algebras over $\mathbb Z/p^n\mathbb Z$" is more general. The field with $4$ elements is a finite algebra over $\mathbb Z/2\mathbb Z$: we can write it as $(\mathbb Z/2\mathbb Z)[x] / (x^2+x+1)$, for instance. But it is not $\mathbb Z/2\mathbb Z$ (or $\mathbb Z/4\mathbb Z$). – Misha Lavrov Oct 12 '20 at 14:12
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    @SunShine The question in the link you gave is every finite commutative ring a product of algebras over $\mathbf{\mathbb Z/p^n}$?" The answers say "yes" to that. They do not say the algebras themselves are $\mathbb Z/(p^n)$. – rschwieb Oct 12 '20 at 14:14

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This is quite false. Actually the classification of finite commutative rings is very complicated. Every such ring is canonically the product $R \cong \prod_p R_p$ of its localizations at each prime $p$; these are "Sylow subrings" of order $p^{\nu_p(|R|)}$. The structure of these can be quite complicated, and in particular there are many more of them than products of $\mathbb{Z}/p^n$. As simple examples, we also have products of finite fields $\mathbb{F}_{p^n}$ or rings like $\mathbb{F}_p[\varepsilon]/\varepsilon^{n+1}$. Asymptotically it's known that there are

$$p^{ \frac{2}{27} n^3 + O \left( n^{ \frac 8 3} \right) }$$

commutative rings of order $p^n$. This is Theorem 11.2 in Bjorn Poonen's The moduli space of commutative algebras of finite rank.

Qiaochu Yuan
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