Isomorphic Simple Modules
Posted: Thu Jun 09, 2016 3:13 pm
- 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?)