Proper divisibility in computable rings

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• 作者：Noam Greenberg ; Alexander Melnikov ; ^{alexander.g.melnuikov@gmail.com}
• 关键词：Constructive (effective) algebra ; Reverse mathematics ; Integral domain theory
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：474
• 期：Complete
• 页码：180-212
• 全文大小：686 K
• 卷排序：474

文摘

We study divisibility in computable integral domains. We develop a technique for coding Σ20 binary trees into the divisibility relation of a computable integral domain. We then use this technique to prove two theorems about non-atomic integral domains.In every atomic integral domain, the divisibility relation is well-founded. We show that this classical theorem is equivalent to ACA0ACA0 over RCA0RCA0.In every computable non-atomic integral domain there is a Δ30 infinite sequence of proper divisions. We show that this upper bound cannot be improved to Δ20 in general.