The Galois closure for rings and some related constructions

Let R be a ring and let A be a finite projective R-algebra of rank n. Manjul Bhargava and Matthew Satriano have recently constructed an R-algebra G(A/R), the Galois closure of A/R. Many natural questions were asked at the end of their paper. Here we address one of these questions, proving the existence of the natural constructions they call intermediate Sn-closures. We will also study properties of these constructions, generalizing some of their results, and proving new results both on the intermediate Sn-closures and on G(A/R).