Abstract
In this paper, we introduce the notion of a free profinite p$p$-ring over a boolean space (over a set). We prove that free profinite p$p$-rings over infinite boolean spaces (over infinite sets) are prime. As a consequence, we obtain that every profinite pro-p$p$-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.
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