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.