# A simple algebra that is not semisimple [duplicate]

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

## marked as duplicate by rschwieb abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jan 22 '17 at 10:18

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.