Ore extensions and infinite triangularization

We give infinite triangularization and strict results for algebras of operators on infinite-dimensional vector spaces. introduce a class we call Ore-solvable algebras: these are similar to iterated Ore extensions but need not be free as modules over the intermediate subrings. include many examples particular cases, such group polycyclic groups or finite solvable groups, enveloping Lie algebras, quantum planes matrices. prove both this class, show how they generalize extend classical simultaneous Engel theorems. that are, in sense, best possible, by showing any finite-dimensional triangularizable algebra must type. also connections between nil nilpotent very general result defined via recursive “Ore” procedure starting from building blocks which either nil, commutative algebras.