On the Krull-dimension of left noetherian left Matlis-rings

Krause, Günter

doi:10.1007/bf01113344

Cited by 32 publications

(13 citation statements)

References 5 publications

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“…0. This expansion is called the normal form of l The Noetherian-dimension of an R-module M is denoted by n-dim M and k-dim M denotes the Krull-dimension of M, see [3,10], for a general background on Krull-dimension of modules.…”

Section: Preliminariesmentioning

confidence: 99%

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On the Loewy Length and the Noetherian Dimension of Artinian Modules

Karamzadeh

Nejad

2002

Communications in Algebra

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Section: Preliminariesmentioning

confidence: 99%

“…We ®rst record some well-known facts, see [3,4,10,6] and [11]. The following is a very useful result, see [3], P 26, Corollary 4.…”

Section: Preliminariesmentioning

confidence: 99%

On the Loewy Length and the Noetherian Dimension of Artinian Modules

Karamzadeh

Nejad

2002

Communications in Algebra

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“…Suppose that U (a) is a Matlis ring. According to [Kra70,Corollary 14], every prime ideal of U (a) is completely prime. Let S be a simple a-module and set D := End F (S).…”

Section: Theorem 13 Let a Be A Finite-dimensional Lie Algebra Over Amentioning

confidence: 99%

Injective Modules and Prime Ideals of Universal Enveloping Algebras

Feldvoss¹

2006

Abelian Groups, Rings, Modules, and Homological Algebra

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In this paper we study injective modules over universal enveloping algebras of finite-dimensional Lie algebras over fields of arbitrary characteristic. Most of our results are dealing with fields of prime characteristic but we also elaborate on some of their analogues for solvable Lie algebras over fields of characteristic zero. It turns out that analogous results in both cases are often quite similar and resemble those familiar from commutative ring theory.

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“…While the Krull dimension and the Gabriel dimension are thus rather well behaved under idealization, the same cannot be said for the classical Krull dimension (cf. [4]), not even in the case of a left noetherian ring. Any simple noetherian integral domain shows that the classical Krull dimension can increase under idealization.…”

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confidence: 99%

Krull Dimension and Gabriel Dimension of Idealizers of Semimaximal Left Ideals

Krause

1976

Journal of the London Mathematical Society

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Let S be a ring with identity, and M a unitary left S-module. If defined, K-dim s (M) and G-dim s (M) denote the Krull dimension and the Gabriel dimension of M, respectively, and K-dim(5") and G-dim(S') stand for the left Krull dimension and the left Gabriel dimension of the ring S.In what follows, we let L always be a semimaximal left ideal (i.e., the intersection of finitely many maximal left ideals) of S; its idealizer in S will always be denoted by R, i.e. R = {s e S \ Ls £ L). In [5] J. C. Robson found the structures of JR and S remarkably closely linked, and it is the purpose of this note to point out a few more similarities. In particular, Robson showed that K-dim(.R) = K-dim(S) if S is left noetherian, and we shall establish this result in general, provided R or S has left Krull dimension. Also, G-dim(R) = G-dim(.S) will be shown, provided either side exists.For some basic information concerning Krull dimension and Gabriel dimension we refer to [2], [3], and [4]. LEMMA 1. Let N be an R-submodule of the left S-module M. If the left S-module M/LN has finite uniform dimension, then N/LN is a semisimple artinian left R-module.

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On the Krull-dimension of left noetherian left Matlis-rings

Cited by 32 publications

References 5 publications

On the Loewy Length and the Noetherian Dimension of Artinian Modules

On the Loewy Length and the Noetherian Dimension of Artinian Modules

Injective Modules and Prime Ideals of Universal Enveloping Algebras

Krull Dimension and Gabriel Dimension of Idealizers of Semimaximal Left Ideals

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