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Math Help - semisimple rings

  1. #1
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    semisimple rings

    If R is a division ring and V is a left vector space over R with dim(V)=n, then End_R(V) is isomorphic to Mat_n(R^{op}) and End_R(V) is a left semisimple ring.

    I already prove the part that End_R(V) is isomorphic to Mat_n(R^{op}), and need help proving that End_R(V) is a left semisimple ring. Any help would be appreciated. Thank you.
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    If R is a division ring and V is a left vector space over R with dim(V)=n, then End_R(V) is isomorphic to Mat_n(R^{op}) and End_R(V) is a left semisimple ring.

    I already prove the part that End_R(V) is isomorphic to Mat_n(R^{op}), and need help proving that End_R(V) is a left semisimple ring. Any help would be appreciated. Thank you.
    a left semisimple ring is also a right semisimple ring and so we just say semisimple. anyway, to prove that if D is a division ring, then Mat_n(D) is semisimple, let I_k, \ 1 \leq k \leq n, be the set of all elements of Mat_n(D) all of whose columns except for the k-th one are zero. it is obvious that Mat_n(D) = I_1 \oplus I_2 \oplus \cdots \oplus I_n. it is also easy to see that each I_k is a minimal left ideal of Mat_n(D). so each I_k is a simple left Mat_n(D) module and we're done.

    we actually have I_k \cong I_{\ell} for all k, \ell and so Mat_n(D) \cong I_1^n. it can also be proved that any simple left Mat_n(D) module is isomorphic to I_1.
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