Birationally commutative graded algebras solve the moduli problem for "point modules" over a graded ring. They have been a fruitful source of counterexamples, examples, and intuition in noncommutative ring theory. We investigate when a large subclass of birationally commutative algebras is noetherian. Formally, these are idealizer subrings of twisted homogeneous coordinate rings. In the process, we give a (purely algebro-geometric) generalization of the Kleiman-Bertini theorem.