Alphabeta MathModules, Submodules, Quotient Modules and the Isomorphism Theorems
About this page
How 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.
✓ 7 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Unital left and right modules over a ring; unqualified module means left module
Lemma
In a module, 0_Rm=0_M, r0_M=0_M, (-r)m=-(rm) and r(-m)=-(rm)
Definition
Submodule of a module
Lemma
The one-step submodule criterion; intersections and sums of submodules are submodules
Definition
Generated submodule, cyclic and finitely generated modules, module basis and free module
Definition
Simple module: a nonzero module with no proper nonzero submodule
Definition
Annihilators, torsion elements and the torsion subset of a module
Definition
Quotient module M/N with scalar multiplication on additive cosets
Definition
Module homomorphism and isomorphism, kernel, image and cokernel
Theorem
Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel
Theorem
The quotient action is well defined and makes M/N a module
Proposition
The canonical map M→ M/N is a surjective module homomorphism with kernel N; thus every submodule is a kernel
Theorem
A module homomorphism vanishing on N factors uniquely through M/N
Theorem