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TitleOn the index of reducibility in Noetherian modules
Author(s) Nguyen Tu Cuong, Pham Hung Quy, Hoang Le Truong
TypeArticle in Journal
AbstractAbstract Let M be a finitely generated module over a Noetherian ring R and N be a submodule. The index of reducibility ir M ( N ) is the number of irreducible submodules that appear in an irredundant irreducible decomposition of N (this number is well defined by a classical result of Emmy Noether). Then the main results of this paper are: (1) ir M ( N ) = ∑ p ∈ Ass R ( M / N ) dim_k ( p ) ⁡ Soc ( M / N ) p ; (2) For an irredundant primary decomposition of N = Q_1 ∩ ⋯ ∩ Q_n , where Q_i is p i -primary, ir M ( N ) = ir M ( Q 1 ) + ⋯ + ir M ( Q n ) if and only if Q i is a p i -maximal embedded component of N for all embedded associated prime ideals p i of N; (3) For an ideal I of R there exists a polynomial Ir M , I ( n ) such that Ir M , I ( n ) = ir M ( I n M ) for n ≫ 0 . Moreover, bight M ( I ) − 1 ≤ deg ⁡ ( Ir M , I ( n ) ) ≤ ℓ M ( I ) − 1 ; (4) If ( R , m ) is local, M is Cohen–Macaulay if and only if there exist an integer l and a parameter ideal q of M contained in m l such that ir M ( q_M ) = dim R / m ⁡ Soc ( H_m^d ( M ) ) , where d = dim ⁡ M .
ISSN0022-4049
URL http://www.sciencedirect.com/science/article/pii/S0022404915000560
LanguageEnglish
JournalJournal of Pure and Applied Algebra
Volume219
Number10
Pages4510 - 4520
Year2015
Edition0
Translation No
Refereed No
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