Maximal subalgebras of central separable algebras

Abstract:

Let A be a central separable algebra over a commutative ring R. A proper R-subalgebra of A is said to be maximal if it is maximal with respect to inclusion. Theorem. Any proper subalgebra of A is contained in a maximal one. Any maximal subalgebra B of A contains a maximal ideal $\mathfrak {m}A$ of A, $\mathfrak {m}$ a maximal ideal of R, and $B/\mathfrak {m}A$ is a maximal subalgebra of the central simple $R/\mathfrak {m}$ algebra $A/\mathfrak {m}A$. More intrinsic characterizations are obtained when R is a Dedekind domain.