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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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    [This might help too](http://ringtheory.herokuapp.com/search/results/?has=7&lacks=8) – 2017-01-22
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    Is not a duplicate since in the other answer there are no examples – 2017-01-22
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    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. – 2017-01-22
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    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. – 2017-01-22

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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.