Ideal invariance and artinian quotient rings

Krause, Günter; Lenagan, T. H.; Stafford, J. T.

doi:10.1016/0021-8693(78)90196-5

Cited by 28 publications

(11 citation statements)

References 9 publications

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“…It follows from [24] that K dim U=2. Since rings of Krull dimension one, enveloping algebras of solvable Lie algebras, integral group rings of polycyclic groups and FBN rings are all weakly ideal invariant (see [9], [22] and [26]) the ring U is about the nicest ring that could possibly not be weakly ideal invariant.…”

Section: P/p M~-(p| @ U)/(u @ U)mmentioning

confidence: 99%

See 1 more Smart Citation

Non-holonomic modules over Weyl algebras and enveloping algebras

Stafford

1985

Invent Math

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Section: P/p M~-(p| @ U)/(u @ U)mmentioning

confidence: 99%

“…A concept that has proved very useful in Noetherian ring theory is that of weak ideal invariance (see, for example, [9], [22] or [33]). This is defined as follows.…”

Section: Weak Ideal Invariancementioning

confidence: 99%

Non-holonomic modules over Weyl algebras and enveloping algebras

Stafford

1985

Invent Math

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“…Suppose that R is a right Noetherian /C-homogeneous ring. In [7] (respectively, [11]) it is shown that R has a right Artinian right quotient ring if N(R) is weakly ideal invariant (respectively, if finitely generated (kdim /? )-critical right /^-modules have prime annihilators).…”

Section: Weak Ideal Invariancementioning

confidence: 99%

Weak Ideal Invariance and Localisation

Brown

Lenagan²,

Stafford

1980

Journal of the London Mathematical Society

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“…We remark that it is an open question whether or not A is a Prime ring if A is supposed to be a Noetherian ring and, moreover, that has a finitely generated, faithful, critical right module M. This question is equivalent to the question of whether or not the nilradical of A is weakly ideal invariant. For this question and related results see [2,7]. We mention that the above question is open even when we assume that A has an Artinian quotient ring.…”

mentioning

confidence: 99%

The structure of some right Noetherian rings with Krull dimension one

Wangneo¹

1993

Proc. Amer. Math. Soc.

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Abstract. In this note we prove a structure theorem for a right Noetherian ring R with krull dimension one having a right Artinian quotient ring and, moreover, that has a finitely generated, faithful, critical right module U with krull dimension U equal to one. We end this note with some examples that clarify certain features of this theorem. Notation and terminologyLet R be a ring and M a right .R-module. We denote the right krull dimension of M or R by \M\ or \R\. For basic properties of krull dimension, the reader is referred to [6]. A module with krull dimension is said to be akrull homogeneous or simply krull homogeneous in the case where \M\ = a, if |TV| = \M\, for every nonzero submodule N of M. The ring R is said to be krull homogeneous if the module Rr is krull homogeneous. For the definition of Primary module or Primary ring and various results about Prime ideals associated to a module we refer the reader to Gordon [4]. If A is a ring then its nilradical will be denoted by N(A). The set of regular elements of A is denoted by C{0) and C(B) denotes the set of elements of A that are regular modulo B where B is an ideal of A . If S,■■, i £ I, are subrings of a ring R then their internal direct sum is denoted by 4-,G/ Sj. A ring R is said to be nonsingular if the right singular ideal of R is zero. The set of associated prime ideals of a right module M is denoted by Ass(M). As usual rings are with identity and modules are unitary.Main theorem Proposition 1. Let R be a right Noetherian ring with M a faithful, finitely generated, critical module over R such that \M\ = \R\. Then R is a krull homogeneous P-primary ring, where P is the associated prime of M. Furthermore, if I is an essential right ideal of R then \R/I\ < \R\.

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Ideal invariance and artinian quotient rings

Cited by 28 publications

References 9 publications

Non-holonomic modules over Weyl algebras and enveloping algebras

Non-holonomic modules over Weyl algebras and enveloping algebras

Weak Ideal Invariance and Localisation

The structure of some right Noetherian rings with Krull dimension one

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