Alphabeta MathHow statement and proof provenance work
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Injective modules and the extension property
Definition
A left -module is injective if it has the extension property for monomorphisms: whenever is an injective module homomorphism and is a module homomorphism, there exists a module homomorphism with (Module homomorphism and isomorphism, kernel, image and cokernel, Injection, surjection, bijection).
The extension need not be unique. Injectivity asks for an extension along every module embedding.
Depends on
Used by
- Over a finite group algebra in defining characteristic, finite-dimensional projective and injective modules coincide Corollary
- Coinduction sends injective abelian groups to injective modules Lemma
- Baer's criterion for injective modules Theorem
- Equivalent characterizations of injective modules Theorem
- Products of injective modules are injective, with the exact choice boundary Theorem
- The character dual of a flat module is injective Theorem
Dependency tree · two levels
6 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
- A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
- The Stacks Project, Algebra (standard reference, not scraped)
- P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)