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.
The endomorphism ring under addition and composition
Definition
For a left -module , define Addition is pointwise and multiplication is composition, . The ring laws and the identity endomorphism are established in Module endomorphisms form a ring under pointwise addition and composition ↗.
Depends on
Used by
- Every surjective endomorphism of a Noetherian module is injective Corollary
- Graded Fitting decomposition for degree-zero endomorphisms Lemma
- Hom_G(V,W) is a k-vector space and End_G(V) is a k-algebra Proposition
- Module endomorphisms form a ring under pointwise addition and composition Proposition
- Endomorphisms of a finite direct sum are matrices of Hom-groups Theorem
- Finite-dimensional kG-modules decompose as finite direct sums of indecomposables uniquely up to order and isomorphism Theorem
- Schur's lemma for simple modules Theorem
Dependency tree · two levels
10 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
- William Crawley-Boevey, Noncommutative Algebra, Chapter 1 Sections 1.1-1.9 (standard reference, not scraped)