Ideals in computable rings

We show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to WKL0 over RCA0, and that the existence of a nontrivial proper finitely generated ideal in a commutative ring with identity which is not a field is equivalent to ACA0 over RCA0. We also prove that there are computable commutative rings with identity where the nilradical is -complete, and the Jacobson radical is -complete, respectively.