The group idempotents in a partial Galois extension

Abstract

Let (R, αG) be a partial Galois extension with a partial action αG of a finite group G, B the Boolean ring generated by {1g | g ∈ G} where 1g is the central idempotent associated with g ∈ G. Let e ≠ 0 ∈ B and G(e) = {g ∈ G | e1g ≠ 0}. We call e a group idempotent if G(e) is a subgroup of G. It is shown that if e is a group idempotent, then (Re, αG(e)) is a partial Galois extension induced by e. Thus the set of these partial Galois extensions in (R, αG) is computed, and a structure theorem for (R, αG) is obtained.