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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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Annihilators, torsion elements and the torsion subset of a module
Definition
For a left -module and , the annihilator of is
and the annihilator of is
If is an integral domain, an element is a torsion element when for some nonzero . The set of all torsion elements is denoted ; is torsion-free when .
Depends on
Used by
- Associated primes of a cyclic quotient are colon primes Corollary
- Every finitely generated torsion-free module over a PID is free Corollary
- Torsion-free abelian groups give a conservative right adjoint that is not monadic Counterexample
- Associated primes of a module Definition
- Primary submodules and primary ideals Definition
- Regular Sequence On A Module Definition
- The p-primary component of a module over a domain Definition
- FALSE: every finitely generated module over a domain is a direct sum of cyclic modules False statement
- FALSE: every torsion-free module over a PID is free False statement
- A cyclic submodule is a residue module by its annihilator Lemma
- A maximal element annihilator is prime Lemma
- A nonzero module over a Noetherian ring has a maximal element annihilator Lemma
- A prime lies in the support exactly when some element has annihilator inside it Lemma
- Associated primes of a localized finite module come from upstairs Lemma
- Torsion elements and p-primary elements form submodules over a domain Proposition
- Assuming the Axiom of Choice, local criteria for zero modules and for injective, surjective, and bijective maps Theorem
- For a finite module, support is the set of primes containing the annihilator Theorem
- Over a principal ideal domain flatness is equivalent to torsion-freeness Theorem
- Torsion-free abelian groups form a reflective full subcategory of abelian groups 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
- McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)