Complexity of some natural problems on the class of computable $I$-algebras

We study computable Boolean algebras with distinguished ideals ($I$-algebras for short). We prove that the isomorphism problem for computable $I$-algebras is $\Sigma_1^1$-complete and show that the computable isomorphism problem and the computable categoricity problem for computable $I$-algebras are $\Sigma_3^0$-complete.