(1) A simple module is semisimple. Vector spaces (over division rings) are semisimple. The ring Z is not a semisimple module over itself.

Is a simple ring semisimple?

A ring is semisimple if and only if it is the ring of endomor- phisms of a finitely generated semisimple module (over some ring). [ss:srings] 9.2. Simple rings. A semisimple ring satisfying any of the equivalent conditions below is called a simple ring.

Are simple modules projective?

Lemma 5.3 Every simple module is a quotient of some cyclic projective indecom- posable module. ... It implies, in particular, that projective indecomposable modules are finitely generated. Lemma 5.4 Every projective indecomposable module is cyclic.