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 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 / R ) . [ABSTRACT FROM AUTHOR]