Primary Decomposition: Compatibility, Independence and Linear Growth
نویسنده
چکیده
For finitely generated modules N ( M over a Noetherian ring R, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if Ass(M/N) = {P1, P2, . . . , Ps} and Qi is a Pi-primary component of N ( M for each i = 1, 2, . . . , s, then N = Q1 ∩Q2 ∩ · · · ∩Qs; (2) For a given subset X = {P1, P2, . . . , Pr} ⊆ Ass(M/N), X is an open subset of Ass(M/N) if and only if the intersections Q1 ∩ Q2 ∩ · · ·∩Qr = Q1∩Q2∩· · ·∩Qr for all possible Pi-primary components Qi and Qi of N ( M ; (3) A new proof of the ‘Linear Growth’ property, which says that for any fixed ideals I1, I2, . . . , It of R there exists a k ∈ N such that for any n1, n2, . . . , nt ∈ N there exists a primary decomposition of I1 1 I n2 2 · · · I nt t M ⊂ M such that every P -primary component Q of that primary decomposition contains P k(n1+n2+···+nt)M .
منابع مشابه
The Compatibility, Independence, and Linear Growth Properties
The first part is about primary decomposition. After reviewing the basic definitions, we survey the compatibility, independence, and linear growth properties that have been known. Then, we prove the linear growth property of primary decomposition for a new family of modules. In the remaining sections, we study secondary representation, which can be viewed as a dual of primary decomposition. Cor...
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