The Logical Complexity of Finitely Generated Commutative Rings
2018 ◽
Vol 2020
(1)
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pp. 112-166
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AbstractWe characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with $(\mathbb{N},{+},{\times })$ if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in $\mathbb{Z}$. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over $\mathbb{Z}$ is not bi-interpretable with $\mathbb{N}$.
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2019 ◽
Vol 13
(07)
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pp. 2050121
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2007 ◽
Vol 06
(04)
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pp. 671-685
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1992 ◽
Vol 111
(1)
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pp. 25-33
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1978 ◽
Vol 21
(3)
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pp. 373-375
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