A ring is not necessarily commutative but is assumed to have the multiplicative identity.

**Proposition.** *Let be a simple ring. Then*

- (i) Every morphism is either zero or an isomorphism. (Schur's lemma)
- (ii)

**Theorem (Levitzky).** *Let be a right noetherian ring. Then every (left or right) nil ideal is nilpotent.*