On the Ideals of a Noetherian Ring

Abstract

Abstract.We construct various examples of Noetherian rings with peculiar ideal structure. For example, there exists a Noetherian domain R with a minimal, nonzero ideal /, such that R/I is a commutative polynomial ring in « variables, and a Noetherian domain S with a (second layer) clique that is not locally finite. The key step in the construction of these rings is to idealize at a right ideal I in a Noetherian domain T such that T/I is not Artinian.

Non-finitely generated projective modules over generalized Weyl algebras

“…Next we will investigate an even more advanced example of Stafford [23]. To keep the notation of his paper, in this example we will consider right modules.…”

Idealizer rings and noncommutative projective geometry

“…Let S be a noetherian ring with left ideal I, and let T = I(I) ⊆ S = {s ∈ S | Is ⊆ I} be the idealizer of I. In [16], Stafford gives a sufficient condition for the left noetherian property of T . In the next proposition, we restate Stafford's result slightly to show that it characterizes the left noetherian property in case S is a finitely generated left T -module, which occurs in many examples of interest.…”

Idealizer rings and noncommutative projective geometry

“…Idealizers have certainly proved useful in the creation of counterexamples before, but it seems that in many natural examples (for example, those in [10] or [14]), the idealizer of a left ideal is a left but not right noetherian ring. Since our intention is to create two-sided noetherian examples, in this brief section we will give some general characterizations of both the left and right noetherian properties for an idealizer ring.…”

“…In [14], Stafford gives a sufficient condition for the left noetherian property of T . In the next proposition, we restate Stafford's result slightly to show that it characterizes the left noetherian property in case S is a finitely generated left T -module, which occurs in many examples of interest.…”