Upper bounds on ideals in the computably enumerable Turing degrees

We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper Σ 4 0 ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no Σ 4 0 prime ideal in the c.e. Turing degrees. This answers a question of Calhoun (1993) [2]. Every proper Σ 3 0 ideal in the c.e. Turing degrees has a low 2 upper bound. Furthermore, the partial order of Σ 3 0 ideals under inclusion is dense.