Idealizer rings and noncommutative projective geometry

We study noetherian graded idealizer rings which have very different behavior on the right and left sides. In particular, we construct noetherian graded algebras T over an algebraically closed field k with the following properties: T is left but not right strongly noetherian; T kT is left but not right noetherian and T kTop is noetherian; the left noncommutative projective scheme T-Proj is different from the right noncommutative projective scheme Proj-T; and T satisfies left χd for some d 2 yet fails right χ1.