Direct Sum Cancellation of Noetherian Modules

Brookfield, Gary

doi:10.1006/jabr.1997.7221

Cited by 16 publications

(9 citation statements)

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“…But A is essential in E, so Y = ker(f ) = 0, as desired. 11 We come now to the main theorem that gives the basic connection between injectivity and exchange. This result was first proved for injective modules by Warfield, and then for quasi-injective modules by Fuchs.…”

Section: Corollary 77 Let a Be A Quasi-injective Module With Injecmentioning

confidence: 99%

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A Crash Course on Stable Range, Cancellation, Substitution and Exchange

Lam

2004

J. Algebra Appl.

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The themes of cancellation, internal cancellation, substitution and exchange have led to a lot of interesting research in the theory of modules over commutative and noncommutative rings. This article provides a quick and relatively self-contained introduction to the voluminous work in this area, using the notion of the stable range of rings as a unifying tool. With only a small number of exceptions, all theorems stated here are proved in full, modulo basic facts in the theory of modules and rings available in standard textbooks on ring theory.

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Section: Corollary 77 Let a Be A Quasi-injective Module With Injecmentioning

confidence: 99%

“…The main result of [11] states that, if A ⊕ B ∼ = A ⊕ C (where A is noetherian), then B and C have isomorphic submodule series, in the sense that there exist submodule series (0) = B 0 B 1 · · · B n = B and (0) = C 0 C 1 · · · C n = C , and a permutation π such that…”

Section: Epiloguementioning

confidence: 99%

A Crash Course on Stable Range, Cancellation, Substitution and Exchange

Lam

2004

J. Algebra Appl.

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“…Denote by R-Noeth the category of all Noetherian left R-modules. As in [6], we say that finite submodule series of same length, say so that we obtain the conclusion for u = c 00 , c = c 01 , v = c 11 .…”

Section: If Smentioning

confidence: 99%

The dimension monoid of a lattice

Wehrung¹

1998

Algebra univers.

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Abstract. We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, ∆, from L × L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Conc L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction:(1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to Z + or to R + . (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power ofis modular, and to the two-element semilattice, otherwise. (5) If L is an ℵ 0 -meet-continuous complemented modular lattice, then both Dim L and the dimension function ∆ satisfy (countable) completeness properties. If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M 2 (R), then Dim L is isomorphic to the monoid V (R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.

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“…In terms of this ordinal valued length, our main cancellation result is the following: This theorem is an interesting contrast to Theorem 1.1(2): Given Noetherian modules A B, and C such that A ⊕ C ∼ = B ⊕ C, Theorem 1.1 (2) guarantees the existence of isomorphic submodule series in A and B but provides no indication of the number of factors, the permutation σ, or the size of the subfactor modules. Theorem 1.2, on the other hand, provides a matchup of submodules of A and B of a specific size.…”

Section: Introductionmentioning

confidence: 96%

“…If, however, we require that C satisfy a chain condition then we get the following two contrasting results: Theorem 1.1. Let A B C be modules such that A ⊕ C ∼ = B ⊕ C. This suggests that the cancellation question is complicated for Noetherian modules, and indeed there are easy examples [2,7] of Noetherian modules A B C such that A ⊕ C ∼ = B ⊕ C, but A ∼ = B.…”

Section: Introductionmentioning

confidence: 99%