1985
DOI: 10.2307/1999707
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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.

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Cited by 9 publications
(9 citation statements)
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“…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.…”
Section: Now We Will Give Some Examples Showing That the Situation Dementioning
confidence: 99%
“…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.…”
Section: Now We Will Give Some Examples Showing That the Situation Dementioning
confidence: 99%
“…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.…”
Section: Idealizer Rings and The Left And Right Noetherian Propertymentioning
confidence: 99%
“…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.…”
Section: Idealizer Rings and The Left And Right Noetherian Propertymentioning
confidence: 99%
“…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.…”
Section: Idealizer Rings and The Left And Right Noetherian Propertymentioning
confidence: 99%