Editing Semisimple module (section)

=== Simple rings ===
{{main|Simple ring}}

One should beware that despite the terminology, ''not all simple rings are semisimple''. The problem is that the ring may be "too big", that is, not (left/right) Artinian. In fact, if ''R'' is a simple ring with a minimal left/right ideal, then ''R'' is semisimple.

Classic examples of simple, but not semisimple, rings are the [[Weyl algebra]]s, such as the '''Q'''-algebra
: <math> A=\mathbf{Q}{\langle x,y \rangle }/\langle xy-yx-1\rangle\ ,</math>
which is a simple noncommutative [[domain (ring theory)|domain]].  These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam's text, in which they are described as nonartinian simple rings.  The [[module theory]] for the Weyl algebras is well studied and differs significantly from that of semisimple rings.