Free profinite p-rings

In this paper, we introduce the notion of a free profinite p-ring over a boolean space (over a set). We prove that free profinite p-rings over infinite boolean spaces (over infinite sets) are prime. As a consequence, we obtain that every profinite pro-p-ring with 1 is a continuous homomorphic image of a profinite prime ring with 1. In addition, we construct examples of profinite prime rings with non-open radical. For a profinite prime ring that satisfies a polynomial identity over its centroid, the radical is open. Furthermore, we prove that every profinite ring of prime characteristic is a subring of a profinite prime ring.