# Questions tagged [semi-simple-rings]

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85 questions
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### An infinite product of fields is not a semisimple ring

I want to show that an infinite product of fields is not a semisimple ring. I know Artin-Wedderburn Theorem, but I wonder can I explain it without using this theorem? Any help will be appreciated. ...
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### Maximal separable subalgebras of semisimple associative algebras

Is anything known about maximal separable subalgebras of semisimple associative (and not necessarily commutative) algebras in finite-dimension? Are those subalgebras of unique dimension or isomorphic ...
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### $(eR)_R$ is a semisimple right $R$-module $\implies$ $eRe$ is a semisimple ring

Let $e$ be an idempotent in a semisimple ring $R$. I want to prove that if $(eR)_R$ is a semisimple right $R$-module, then $eRe$ is a semisimple ring. The things I have done so far: Since $(eR)_R$ ...
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### Is $\mathrm{End}_{KS_r}(V)$ semisimple?

If $K$ is a field of characteristic $0$, then the ring $KS_r$ is semisimple. Let $V$ be a $KS_r$-module. In particular it is semisimple. Can we assure that $\mathrm{End}_{KS_r}(V)$ is a semisimple ...
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### (Ex 3.13 Lam) Suppose R is a simple, infinite dimensional algebra over a field k. If V is left R module, then V as vsp over k is infinite dimensional.

My approach: Since R is infinite dimensional over k, therefore the infinite basis are $(r_1,...r_i..)$. Let $r_i.v = w_i\in V$ (Since $V$ is a left $R$ module). Now I prove that $w_i$ are independent. ...
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### (Ex 3.9 b, Lam) If $R$ is semisimple ring with $aR = I$ a two-side ideal, then prove that $Ra = I$.

My approach: Suppose I know that $aR = R$; $a \in R$, $\implies Ra = R$ for a semisimple ring $R$ Now I think I can apply this result as follows. Since $I$ is an ideal in $R$, it can be considered as ...