If S is a simple module and f : S → T is a module homomorphism, then f is either zero or injective. (Reason: the kernel of f is a submodule of S and hence is either 0 or S.) If T is also simple, then f is either zero or an isomorphism. (Reason: the image of f is a submodule of T and hence either 0 or T.) Taken together, this implies that the endomorphism ring of a simple module is a division ring.
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