This note presents a Galois
theory for a separable algebra Λ over a semi-local ring R without imposing any
restrictions on the presence of idempotent elements in Λ. If suitable restrictions are
placed on the existence of idempotent elements in Λ, then such a Galois
theory has been obtained by L. N. Childs and F. R. DeMeyer; and the results
presented here are extensions of results obtained by Childs and DeMeyer.
In more recent work, DeMeyer has extended the Galois theory to algebras
over a class of commutative rings more general than semi-local rings. After
treating the simpler case of algebras over a semi-local ring, it is indicated
in an addendum to this paper how the Galois theory presented here may
also be extended to algebras over this more general class of commutative
rings.
