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Schur's lemma for simple modules
Statement
A nonzero homomorphism between simple modules is an isomorphism. Consequently the endomorphism ring of a simple module is a division ring. See Simple module: a nonzero module with no proper nonzero submodule.
Facts & Assumptions
Given: The hypotheses and objects in the Statement.
A left -module is simple if and its only submodules are and . Equivalently, has no proper nonzero submodule. (Simple module: a nonzero module with no proper nonzero submodule).
For a left -module , define Addition is pointwise and multiplication is composition, . The ring laws and the identity endomorphism are established in prop-endomorphisms-form-a-ring. (The endomorphism ring under addition and composition).
For every left -module , pointwise addition and composition make a unital ring with identity . (Module endomorphisms form a ring under pointwise addition and composition).
Proof
The kernel and image of a homomorphism between simple modules are each zero or whole.
A nonzero homomorphism is therefore injective and surjective.
Applied to a nonzero endomorphism, its inverse is linear, so the endomorphism ring is a division ring.
The excluded case is genuinely excluded rather than overlooked: the zero homomorphism between nonzero simple modules is not an isomorphism, which is why the hypothesis asks for a nonzero one, and it is the zero element of the endomorphism ring of step 3.1 rather than a non-invertible unit. Contrapositively, if two simple modules are not isomorphic then every homomorphism between them is zero. This proves the stated claim.
Depends on
Used by
- Schur’s lemma for irreducible Lie-algebra representations Corollary
- Schur's lemma for irreducible representations: a nonzero intertwiner is an isomorphism, and End_G(V) is a division ring Corollary
- A finite-dimensional algebra separates its split simple modules Lemma
- Projective and simple classes are dual bases under splitting Theorem
- Simple modules over a product of matrix rings over division rings Theorem
- Uniqueness of the Wedderburn–Artin factors Theorem
- Wedderburn–Artin theorem for semisimple rings 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
- William Crawley-Boevey, Noncommutative Algebra, Chapter 1 Sections 1.1-1.9 (standard reference, not scraped)