The Logical Complexity of Finitely Generated Commutative Rings
The Logical Complexity of Finitely Generated Commutative Rings
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有限生成交换环的逻辑复杂性
DOI：
10.1093/imrn/rny023
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发表时间：
2018
影响因子：
1
通讯作者：
Scanlon, Thomas
中科院分区：
文献类型：
--
作者：
Aschenbrenner, Matthias;Khélif, Anatole;Naziazeno, Eudes;Scanlon, Thomas
我们描述了那些可以用算术（参数）双向解释的有限生成交换环：当且仅当 A 的非极大素理想的空间非空且在 Zariski 拓扑中连通，并且 A 的零根有一个非平凡消灭子时，有限生成交换环 A 是可以双向解释的。值得注意的是，通过在非标准算术模型上构造非平凡推导，我们证明了双数环不可双向解释。
We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ringAis bi-interpretable withif and only if the space of non-maximal prime ideals ofAis nonempty and connected in the Zariski topology and the nilradical ofAhas a nontrivial annihilator in. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers overisnotbi-interpretable with.