Topics in Abstract Algebra/Non-commutative rings
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)
is either zero or an isomorphism. (Schur's lemma)Theorem (Levitzky). Let be a right noetherian ring. Then every (left or right) nil ideal is nilpotent.