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 $G(A\mathrm{/}R)$, the Galois closure of $A\mathrm{/}R$. 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 $S_n$-closures. We will also study properties of these constructions, generalizing some of their results, and proving new results both on the intermediate $S_n$-closures and on $G(A\mathrm{/}R)$.