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This question already has an answer here:

I was said that can exists simple algebras $A$ that are not semi-simples in the following sense:

  1. $A$ is simple, i.e. doesn't have non trivial ideals;
  2. $A$ is not semi-simple, i.e. is not a semi-simple module over itself.

Does anybody has an example? Thanks in advance

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marked as duplicate by rschwieb abstract-algebra Jan 22 '17 at 10:18

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ This might help too $\endgroup$ – rschwieb Jan 22 '17 at 10:23
  • $\begingroup$ Is not a duplicate since in the other answer there are no examples $\endgroup$ – Dac0 Jan 22 '17 at 11:34
  • $\begingroup$ you apparently did not read it then, because the linked question gives Weyl algebras as an example. When I saw your comment about wanting more examples, I added the link above. $\endgroup$ – rschwieb Jan 22 '17 at 11:36
  • $\begingroup$ I don't see why you would close the question saying it's a duplicate while it is not since my question is just about explicit and concrete examples while the answer you pointed out it's just a theoretical justification. Anyway your link is very intersting thank you. $\endgroup$ – Dac0 Jan 22 '17 at 13:51
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Yes, the Weyl algebra $A = \mathbb{C}[x, \partial]/(\partial x - x \partial - 1)$ is a standard example. Proving that it's simple but not semisimple is a nice exercise.

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