Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Primary submodules and primary ideals
Definition
Let be a commutative ring, let be a left -module, and let be a proper submodule. Then is primary when every zero divisor on the quotient module acts nilpotently on ; equivalently, whenever multiplication by on has nontrivial kernel, there exists with
If
then is called -primary.
The radical here is well-defined: is an ideal of . Indeed, it contains , is closed under subtraction, and if annihilates then so does for every , by the module axioms.
When , viewed as its regular module, a primary submodule is a primary ideal.
Depends on
Used by
- Primary decompositions, minimality, and isolated components Definition
- In a concrete Artinian local quotient, maximal radical forces primaryity Example
- Localizing (x²,xy)=(x)∩(x,y)² keeps only the matching component Example
- The zero module has empty support and no associated primes Example
- A finite intersection of primary submodules with one radical is primary Lemma
- Primary submodules are exactly quotients with nilpotent zero divisors Lemma
- Primary submodules of finite modules are characterized by a singleton associated-prime set Theorem
- The radical of a primary ideal is prime Theorem
Dependency tree · two levels
11 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
- J. S. Milne, A Primer of Commutative Algebra, v4.03, Definition 19.1 (standard reference, not scraped)
- A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §18 (standard reference, not scraped)