On the lattice of subalgebras of an algebra

Abstract:

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