Details:
Title  On the index of reducibility in Noetherian modules  Author(s)  Nguyen Tu Cuong, Pham Hung Quy, Hoang Le Truong  Type  Article in Journal  Abstract  Abstract 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 .  ISSN  00224049 
URL 
http://www.sciencedirect.com/science/article/pii/S0022404915000560 
Language  English  Journal  Journal of Pure and Applied Algebra  Volume  219  Number  10  Pages  4510  4520  Year  2015  Edition  0  Translation 
No  Refereed 
No 
