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Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.

If one replaces the underlying field by a ring, and therefore looks at modules, things become more tricky. Many pathologies occur that one maybe would not expect coming from linear algebra.

I am looking for a list of such pathologies where modules behave differently than vector spaces. This list should not only be a list of statements but all phenomena should be illustrated by an example.

To start the list I will post an answer below with all pathologies that I know from the top of my head. This should also explain better what kind of list I have in mind.

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    See http://math.stackexchange.com/questions/196203/an-indecomposable-mathbfz-module-whose-injective-hull-is-not-indecomposable for an example where two notions of dimension give answers of 1 and uncountably-infinite for the same module.2012-09-24
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    Should this be CW?2012-09-24
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    @kahen: I thought I did this already when posting the question. Now I cannot find the checkbox anymore. How do I make it community wiki?2012-09-24
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    I would suggest every write **one** pathology in each answer.2012-09-24
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    @FredrikMeyer What would the utility be? There would be at least thirteen answers now...2012-09-24

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