On the lattice of subalgebras of an algebra

Abstract

Let R R be a Noetherian inertial coefficient ring and let A A be a finitely generated R R -algebra (that is, finitely generated as an R R -module) with Jacobson radical J ( A ) J(A) . Let S S be a subalgebra of A A with S + J ( A ) = A S + J(A) = A . We show that for every separable subalgebra T T of a a there is a unit a of A A such that a T a − 1 ⊆ S aT{a^{ - 1}} \subseteq S . It follows that if S S is separable (hence inertial) and if T T is a maximal separable subalgebra of A A , then T T is inertial. We also show that if S + I = A S + I = A for a nil ideal I I of A A , then R R can be taken to be an arbitrary commutative ring, and the conjugacy result still holds.