Rings and Categories of Modules

Anderson, Frank W.; Fuller, Kent R.

doi:10.1007/978-1-4612-4418-9

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Preliminaries and Notations1

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1994

2021

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Cited by 871 publications

(321 citation statements)

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“…For this purpose, some finiteness assumptions are required. (1). Assume that A has right almost split sequences.…”

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confidence: 99%

Almost Split Sequences and Approximations

Liu

Paquette

2012

Algebr Represent Theor

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Abstract. Let A be an exact category, that is, an extension-closed full subcategory of an abelian category. First, we give new characterizations of an almost split sequence in A, which yields some necessary and sufficient conditions for A to have an almost split sequence with prescribed end terms. Then, we study when an almost split sequence in A induces an almost split sequence in an exact subcategory C of A. In case A has almost split sequences and C is Ext-finite and Krull-Schmidt, we obtain a necessary and sufficient condition for C to have almost split sequences. Finally, we show some applications of these results.

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“…For this purpose, some finiteness assumptions are required. (1). Assume that A has right almost split sequences.…”

mentioning

confidence: 99%

Almost Split Sequences and Approximations

Liu

Paquette

2012

Algebr Represent Theor

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“…, v t (y) form k-bases of ker B(y), for all y ∈ V x . As a consequence of the previous considerations, we infer that v (1) j (x) = 0 for some 1 ≤ j ≤ t. But then, setting U x := {y ∈ V x ; v (1) j (y) = 0} we get that v (1) (2) j (y) and ρ i depends regularly on y ∈ U x . Finally, it is clear, that the inequalities ρ i (x) = 0, for x ∈ X, imply that ρ i ∈ k[X] * for every i = 1, .…”

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confidence: 67%

Degenerations of Algebras and Ranks of Grothendieck Groups

Hajduk

2012

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“…(For the terminology and notations used here we refer to [1], [2].) Throughout this paper, all rings are associative with identity, and all modules are left unitary modules.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%