Schur lemma says that a homomorphism between simple modules is either zero or an isomorphism; consequently the endomorphism ring of a simple module is a division ring. Indeed, the kernel and image of a module homomorphism are submodules. Simplicity makes each either zero or the whole module, proving both assertions.
After replacing by a maximal -linearly independent subset with the same span, write . The standard independence lemma, proved by induction using Schur's lemma, saysOtherwise would depend only on , defining an -map whose coordinate maps lie in and forcing . Applying the same argument with shows that implies , proving the requested claim.
The Jacobson density theorem states that if are -independent and , there is with for every . Induct on . First match the first values. The independence lemma makes , for , a nonzero submodule and hence all of ; an element of supplies the final correction.
If is primitive, choose a faithful simple module . When , density makes surjective and faithfulness makes it injective. If is infinite, choose an -dimensional -subspace and let . Density makes restriction surjective.
For a simple -module , matrix units show that every nonzero vector of generates all coordinates, so is a simple -module. Conversely, if is simple over the matrix ring, is a simple -module andthrough the maps induced by and . These constructions are inverse on isomorphism classes; this is the basic Morita equivalence for a matrix ring.
The Jacobson radical is the intersection of annihilators of all simple left modules. On , a matrix annihilates every vector exactly when each entry annihilates . Intersecting over all simple gives
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