On the complexity of radicals in noncommutative rings

This article expands upon the recent work by Downey et al. (2007) [3], who classified the complexity of the nilradical and Jacobson radical of commutative rings in terms of the arithmetical hierarchy.Let R be a computable (not necessarily commutative) ring with identity. Then it follows from the definitions that the prime radical of R is , and the Levitzki radical of R is . We show that these upper bounds for the complexity of the prime and Levitzki radicals are optimal by constructing two noncommutative computable rings with identity, such that the prime radical of one is -complete, while the Levitzki radical of the other is -complete.