semisimple rings

**dori1123** · July 2nd 2010, 02:07 PM

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.

**NonCommAlg** · July 2nd 2010, 05:05 PM

Originally Posted by dori1123

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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