The structure of a partial Galois extension

Abstract

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Partial Galois cohomology and related homomorphisms

“…We conclude that J g = D g u g , u g is a free generator of J g over D g and u g r = α g (r1 g −1 )u g , for all r ∈ R, g ∈ G, in view of (38).…”

“…Let f be an element of Z 1 (G, α * , PicS(R)) and write f (g) = [M g ]. Then J g = M g ⊗ g (D g −1 ) I and (38) x g r = α g (r1 g −1 )x g , for any x g ∈ J g , r ∈ R, and the map…”

Galois extensions induced by a central idempotent in a partial Galois extension

“…Galois theory for fields has been generalized for rings in [1,3,4,8]. Recently, a partial action on a ring of a finite group had many applications in operator algebra, ring theory and other areas of research [2,6,7,9,10]. A lot of properties of a partial Galois extension of a ring with a partial action of a finite group have been given ( [6,9]).…”

“…Recently, a partial action on a ring of a finite group had many applications in operator algebra, ring theory and other areas of research [2,6,7,9,10]. A lot of properties of a partial Galois extension of a ring with a partial action of a finite group have been given ( [6,9]). Let (R, α G ) be a partial Galois extension with a partial action of a finite group G. Denote the Boolean semi-group generated by {1 g |g ∈ G} under the multiplication of R by B(R), where 1 g is the central idempotent associated with g ∈ G. In [9], a Galois extension Rf is characterized for an f ∈ B(R).…”

“…A lot of properties of a partial Galois extension of a ring with a partial action of a finite group have been given ( [6,9]). Let (R, α G ) be a partial Galois extension with a partial action of a finite group G. Denote the Boolean semi-group generated by {1 g |g ∈ G} under the multiplication of R by B(R), where 1 g is the central idempotent associated with g ∈ G. In [9], a Galois extension Rf is characterized for an f ∈ B(R). For any non-zero central idempotent e ∈ R, not necessarily in B(R), the purpose of the present paper is to give a sufficient condition for three subsets G(e), N (e), and H(e) of G induced by e to be subgroups of G respectively, where G(e) = {g ∈ G|e1 g = 0}, N (e) = {g ∈ G|e(Π g∈G 1 g ) = 0} with a maximal number of factors 1 g and H(e) = {g ∈ G|e1 g = e}.…”