2014/05/06 by Nguyen Tu Cuong, Cuong, Nguyen Tu, Phạm Hùng Quý +4
Computer Science · Mathematics · #13C99 #13H10 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Primary 13A15 #Secondary 13D45 #math.AC #msc:13A15 #msc:13C99 #msc:13D45 #msc:13H10
paper · pdf · doi:10.48550/arxiv.1405.1136
14 pages, To appear in J. Pure Appl. Algebra
openalex publication_date 2014/05/06 · arxiv created 2015/04/10 · arxiv updated 2015/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a finitely generated module over a Noetherian ring R and N a submodule. The index of reducibility irM(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) irM(N) = ∑_\frak p ∈ AssR(M/N) dimk(\frak p) Soc(M/N)\frak p ; (2) For an irredundant primary decomposition of N = Q1 ∩ ⋯ ∩ Qn, where Qi is \frak pi-primary, then irM(N) = irM(Q1) + ⋯ + irM(Qn) if and only if Qi is a \frak pi-maximal embedded component of N for all embedded associated prime ideals \frak pi of N; (3) For an ideal I of R there exists a polynomial IrM,I(n) such that IrM,I(n)=irM(InM) for n≫ 0. Moreover, bightM(I)-1≤ °(IrM,I(n))≤ ℓM(I)-1; (4) If (R, \frak m) is local, M is Cohen-Macaulay if and only if there exist an integer l and a parameter ideal \frak q of M contained in \frak ml such that irM(\frak qM)=dimkSoc(Hd\frak m(M)), where d=dim M.