A question of Guralnick about reflexive algebras (Q1345500)

Let \(A\) be a commutative, finite-dimensional, local \(k\)-algebra, and set \(B = \text{End}_A (M)\), where \(M\) is a finitely generated, faithful, indecomposable \(A\)-module. \textit{Guralnick} asked whether \(B\) must equal \(A\) if \(B\) is an intermediate algebra of \(A\), i.e., if the \(A\)- and \(B\)- submodules of \(M\) are the same. We give a class of examples where this is not the case.