Boolean algebras of elementary characteristic (1,0,1) whose set of atoms and Ershov–Tarski ideal are computable

It is proved that there exists a computable Boolean algebra of elementary characteristics (1,0,1) which has a computable set of atoms and a computable Ershov–Tarski ideal, but no strongly computable isomorphic copy. Also a description of $\Delta^0_6$-computable Boolean algebras is presented.