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Monthly Archives: December 2014
Structure of Semisimple Rings
It turns out there is a nice classification for semisimple rings. Theorem. Any semisimple ring R is a finite product: where each is a division ring and is the ring of n × n matrices with entries in D. Furthermore, the … Continue reading
Semisimple Rings and Modules
After discussing simple modules, the next best thing is to look at semisimple modules, which are just direct sums of simple modules. Here’s a summary of the results we’ll prove: A module is semisimple iff it is a sum of simple … Continue reading
Simple Modules
We briefly talked about modules over a (possibly non-commutative) ring R. An important aspect of modules is that unlike vector spaces, modules are usually not free, i.e. they don’t have a basis. For example, take the Z-module given by Z/2Z. [ Recall: a Z-module is … Continue reading
Posted in Notes
Tagged algebra, associative algebra, division rings, rings, schur's lemma, simple modules
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Coming up next…
This blog has been dormant for a while, as I’ve been doing quite a bit of self-reading and ruminating over the stuffs I’ve read. I’d really like to post some of my thoughts, but there’s always the risk of misleading … Continue reading
Posted in Uncategorized
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Mathematics and Such 