Isomorphic Simple Modules

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

Isomorphic Simple Modules

#1

Post by Tsakanikas Nickos »

  • Let \( \displaystyle D \) be a division ring and let \( \displaystyle R = \mathbb{M}_{n}(D) \) be the ring of \( \displaystyle n \times n \) matrices over \( \displaystyle D \). Show that \( \displaystyle R \) has a unique (up to isomorphism) simple left \( \displaystyle R \)-module.
  • Let \( \displaystyle \left\{ D_{j} \right\} _{j=1}^{k} \) be division rings and let \( \displaystyle \left\{ \mathbb{M}_{n_{j}}(D_{j}) \right\} _{j=1}^{k} \) be the respective rings of \( \displaystyle n_{j} \times n_{j} \) matrices over \( \displaystyle D_{j} \). Furthermore, let \( \displaystyle S = \prod_{j=1}^{k} \mathbb{M}_{n_{j}}(D_{j}) \) be their direct product. Can the above be generalised for \( \displaystyle S \)? And if so, how? (In other words, how many isomorphism classes of simple left \( \displaystyle S \)-modules exist?)
Papapetros Vaggelis
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Joined: Mon Nov 09, 2015 1:52 pm

Re: Isomorphic Simple Modules

#2

Post by Papapetros Vaggelis »

It's known from \(\displaystyle{\rm{Wedderburn}}\) theorem, that if an associative ring with unity

is semesimple, then \(\displaystyle{R\cong \prod_{i=1}^{s}\mathbb{M}_{n_{i}}(D_{i})}\) as rings,

where \(\displaystyle{n_i\in\mathbb{N}}\) and \(\displaystyle{D_{i}}\) are division rings. The number

\(\displaystyle{s}\) measures the simple \(\displaystyle{R}\) - modules, which, are not isomorphic.

So, the answer in our case is \(\displaystyle{1}\) and \(\displaystyle{k}\), respectively.
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