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Primary decompositions, minimality, and isolated components
Definition
Let be a commutative ring, let be a left -module, and let be a submodule.
A primary decomposition of in is an expression
with each a primary submodule.
Such a decomposition is minimal when:
- no component is redundant, so omitting any changes the intersection;
- the radicals are pairwise distinct.
These radicals are well-defined because Primary submodules and primary ideals establishes that each module annihilator is an ideal of .
In a minimal decomposition, a component is isolated when its radical is minimal, under inclusion, among the radicals occurring in the decomposition.
Depends on
Used by
- Localizing (x²,xy)=(x)∩(x,y)² keeps only the matching component Example
- A finite primary decomposition can be stripped of redundant components Lemma
- Equal-radical primary components can be combined Lemma
- The radicals in a minimal primary decomposition are exactly the associated primes of the quotient Lemma
- Isolated primary components are recovered by localization and contraction Theorem
Dependency tree · two levels
8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., (18.13) (standard reference, not scraped)
- J. S. Milne, A Primer of Commutative Algebra, v4.03, Definition 19.7 (standard reference, not scraped)