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I have wondered the decidability of elementary theory of finite commutative rings. Since we know that the elementary theory of finite fields is decidable shown by J.Ax (The Elementary Theory of Finite Fields,1968) but I cannot found the results of finite commutative rings.

The most related article is Elementary theory of a finitely generated commutative ring shown by G.A.Noskov which says that the elementary theory of a finitely generated commutative ring is undecidable. However, this paper is about a single finitely generated commutative ring rather than a class of finite commutative rings.

So is the elementary theory of finite commutative rings decidable or undecidable?

Any comments or answers are welcomed. Thank you!

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