Krull dimension-nilpotency and Gabriel dimension

Gordon, Robert; Lenagan, T. H.; Robson, J. C.

doi:10.1090/s0002-9904-1973-13277-x

Cited by 37 publications

(33 citation statements)

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“…7]) that every noetherian module has Krull dimension. It was also shown by Krause [Kr] and Michler [Mi] that a module with Krull dimension has finite uniform dimension (see [GR,Proposition 1.4,p. 7]).…”

Section: ])mentioning

confidence: 93%

“…If there is no ordinal α such that Kdim M = α, we say that M has no Krull dimension (see [GR,p. 5] or [Br]).…”

Section: ])mentioning

confidence: 99%

“…If a module contains no infinite direct sum of nonzero submodules, then there is a fixed bound on the number of summands in finite sums it can contain; and the module is said to have finite uniform dimension (see [GR,p. 7]).…”

Section: ])mentioning

confidence: 99%

“…It was proved by Gabriel ([Ga], see also [GR,Proposition 1.3,p. 7]) that every noetherian module has Krull dimension.…”

Section: ])mentioning

confidence: 99%

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Polyhedra with virtually polycyclic fundamental groups have finite depth

Kołodziejczyk

2007

Fund. Math.

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Abstract. The notions of capacity and depth of compacta were introduced by K. Borsuk in the seventies together with some open questions. In a previous paper, in connection with one of them, we proved that there exist polyhedra with polycyclic fundamental groups and infinite capacity, i.e. dominating infinitely many different homotopy types (or equivalently, shapes). In this paper we show that every polyhedron with virtually polycyclic fundamental group has finite depth, i.e., there is a bound on the lengths of all descending sequences of different homotopy types (or shapes) dominated by this polyhedron. As a corollary, we deduce that for two ANR's with virtually polycyclic fundamental groups the so-called index of h-proximity, introduced by K. Borsuk in his monograph on retract theory, is finite. We also obtain an answer to some question of K. Borsuk concerning homotopy (or shape) decompositions of polyhedra into simple constituents.

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Section: ])mentioning

confidence: 93%

“…If there is no ordinal α such that Kdim M = α, we say that M has no Krull dimension (see [GR,p. 5] or [Br]).…”

Section: ])mentioning

confidence: 99%

Section: ])mentioning

confidence: 99%

“…It was proved by Gabriel ([Ga], see also [GR,Proposition 1.3,p. 7]) that every noetherian module has Krull dimension.…”

Section: ])mentioning

confidence: 99%

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Polyhedra with virtually polycyclic fundamental groups have finite depth

Kołodziejczyk

2007

Fund. Math.

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“…If R has Gabriel dimension, so does every left 7?-module. For the relevant definitions and more information on the Gabriel dimension, see [8] or [9].…”

Section: Mark L Teplymentioning

confidence: 99%

On the idempotence and stability of kernel functors

Teply

1995

Glasgow Math. J.

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The Injective Spectrum of a Noncommutative Space

Pappacena¹

2002

Journal of Algebra

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For a noncommutative space X, we study Inj X , the set of isomorphism classes of indecomposable injective X-modules. In particular, we look at how this set, suitably topologized, can be viewed as an underlying "spectrum" for X. As applications we discuss noncommutative notions of irreducibility and integrality, and a way of associating an integral subspace of X to each element of Inj X which behaves like a "weak point."  2002 Elsevier Science (USA)

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Krull dimension-nilpotency and Gabriel dimension

Cited by 37 publications

References 9 publications

Polyhedra with virtually polycyclic fundamental groups have finite depth

Polyhedra with virtually polycyclic fundamental groups have finite depth

On the idempotence and stability of kernel functors

The Injective Spectrum of a Noncommutative Space

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