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Thread: semisimple rings

  1. #1
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    semisimple rings

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

    I already prove the part that $\displaystyle End_R(V)$ is isomorphic to $\displaystyle Mat_n(R^{op})$, and need help proving that $\displaystyle End_R(V)$ 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 $\displaystyle R$ is a division ring and $\displaystyle V$ is a left vector space over $\displaystyle R$ with $\displaystyle dim(V)=n$, then $\displaystyle End_R(V)$ is isomorphic to $\displaystyle Mat_n(R^{op})$ and $\displaystyle End_R(V)$ is a left semisimple ring.

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

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