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: every morphism $R\to R$ is either zero or an isomorphism. (Schur's lemma)

Theorem (Levitzky).

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