14 - Rings Satisfying the Second Layer Condition

Summary

The main theme of this chapter is the exploration of the ideal theory of noetherian rings satisfying the second layer condition. This is a very large class of rings (as we began to see in the previous chapters), including many iterated differential operator rings, iterated skew-Laurent extensions, and quantized coordinate rings, as well as the group rings of polycyclic-by-finite groups and the enveloping algebras of finite dimensional solvable Lie algebras. It turns out that these rings have many properties that are not shared by other noetherian rings and that can be thought of as generalizations of well-known properties of commutative rings. We begin with a symmetry property of bimodules over these rings. This will give us immediate information about the graphs of links of these rings and will also give us the key tool to prove two intersection theorems – a strong form of Jacobson's Conjecture and an analogue of the Krull Intersection Theorem. Rings satisfying the second layer condition also behave well with respect to finite extensions. If R is a noetherian ring satisfying the second layer condition, and R is a subring of a ring S such that S is finitely generated as both a left and a right R-module, we prove that S also satisfies the second layer condition, and that “Lying Over” holds for the prime ideals in this setup.