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