Tag Archives: associative algebra

Local Rings

Mathematicians are generally more familiar with the case of local commutative rings, so we’ll begin from there. Definition. A commutative ring R is said to be local if it has a unique maximal ideal. Note that every non-zero commutative ring has … Continue reading

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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

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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

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