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Indecomposables live in all smaller lengths
Author(s) -
Ringel Claus Michael
Publication year - 2011
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdq128
Subject(s) - indecomposable module , mathematics , simple (philosophy) , simple module , assertion , pure mathematics , algebra over a field , combinatorics , computer science , programming language , philosophy , epistemology
Let Λ be a finite‐dimensional k ‐algebra with k algebraically closed. Bongartz has recently shown that the existence of an indecomposable Λ‐module of length n > 1 implies that also indecomposable Λ‐modules of length n − 1 exist. Using a slight modification of his arguments, we strengthen the assertion as follows: If there is an indecomposable module of length n , then there is also an accessible one. Here, the accessible modules are defined inductively, as follows. First, the simple modules are accessible. Second, a module of length n ⩾ 2 is accessible provided it is indecomposable and that there is a submodule or a factor module of length n − 1 which is accessible.