The Complexity of Nilradicals and Jacobson Radicals in Computable Rings

Studies in Logic, Vol. 15, No. 3 (2022): 36–51 PII: 1674-3202(2022)-03-0036-16

 Xun Wang

Abstract. This paper expands upon the work by Downey et al. (2007), who proved that there are computable commutative rings with identity where the nilradical is -complete, and the Jacobson radical is-complete, respectively. We simplify the proof, showing that there is a computable commutative ring with identity where the nilradical is -complete and meanwhile the Jacobson radical is -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 any set B there exists a computable commutative ring with identity where the Jacobson radical is Turing equivalent to B.