On constructing orthogonal idempotents. (Q2478023)

Let \(A\) be a commutative semisimple algebra over an algebraically closed field. Let \(e_1,\dots,e_{n-1}\), where \(3\leq n\leq\dim A\), be nontrivial orthogonal idempotents in \(A\) such that \(e_1,\dots,e_{n-2}\) are minimal. It is shown that then there exist nontrivial orthogonal idempotents \(q_1,\dots,q_n\) with \(q_1,\dots,q_{n-1}\) minimal.