可计算环上幂零根与Jacobson根的计算复杂度  

The Complexity of Nilradicals and Jacobson Radicals in Computable Rings

作　　者：王勋 Xun Wang(Department of Philosophy,Peking University)

机构地区：[1]北京大学哲学系

出　　处：《逻辑学研究》2022年第3期36-51,共16页Studies in Logic

摘　　要：Downey等人(2007)证明了:存在一个可计算的有单位元素的交换环,其幂零根是Σ_(1)^(0)-完全集;存在另一个可计算的有单位元素的交换环,其Jacobson根是Π_(2)^(0)-完全集。本文进一步证明了:存在一个可计算的有单位元素的交换环,其幂零根是Σ_(1)^(0)-完全集且其Jacobson根是Π_(2)^(0)-完全集。此外,对于任意c.e.集A,都存在一个可计算的有单位元素的交换环使其幂零根与A图灵等价;对于任意Π_(2)^(0)集B,都存在一个可计算的有单位元素的交换环使其Jacobson根与B图灵等价。This paper expands upon the work by Downey et al.(2007),who proved that there are computable commutative rings with identity where the nilradical isΣ_(1)^(0)-complete,and the Jacobson radical isΠ_(2)^(0)-complete,respectively.We simplify the proof,showing that there is a computable commutative ring with identity where the nilradical isΣ_(1)^(0)-complete and meanwhile the Jacobson radical isΠ_(2)^(0)-complete.Moreover,we show that for any c.e.set A there exists a computable commutative ring with identity where the nilradical is Turing equivalent to A,and for anyΠ_(2)^(0)-set B there exists a computable commutative ring with identity where the Jacobson radical is Turing equivalent to B.

关 键 词：JACOBSON根 交换环 幂零根 计算复杂度 

分 类 号：O153.3[理学—数学]