Topics in Abstract Algebra/Non-commutative rings

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

Proposition. Let $R$ be a simple ring. Then

• (i) Every morphism $R \to R$ is either zero or an isomorphism. (Schur's lemma)
• (ii)
  

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