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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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    $\begingroup$ According to page 220 of Provability, Computability and Reflection By Lev D. Beklemishev the elementary theory of all finite rings or all finite rings of a given prime characteristic is undecidable, but I believe that he is talking about non-commutative rings (I only saw the google preview). $\endgroup$
    – Benjamin Steinberg
    Commented Feb 5, 2018 at 21:17
  • $\begingroup$ @BenjaminSteinberg Thank you! I think this books is The metamathematics of algebraic systems which is the collected papers of A.I.Mal'cev during 1936 and 1967. What A.I.Mal'cev proved in P.220 is the elementary theory of all finite rings which is not necessarily commutative and associative. (Quite general!). $\endgroup$
    – Max CYLin
    Commented Feb 6, 2018 at 16:52

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