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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- 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.
Composition series and length of a module
Definition
A composition series of a left -module is a finite chain whose factors are simple. If such a series exists, the length is its number of factors; Jordan–Hölder theorem for modules ↗ proves independence of the chosen series. The zero module has the empty series and length .
Depends on
Used by
- Module length is additive in short exact sequences Corollary
- The ℤ-module ℤ/pᵏ has length k Example
- A module has a composition series if and only if it is Noetherian and Artinian, the converse using dependent choice Theorem
- Choice-free semisimple characterizations for finite-length modules Theorem
- Jordan–Hölder theorem for modules Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 14 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Arvind Nair, Algebra I, Lecture 5 (standard reference, not scraped)