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

Proposition. Let ${\displaystyle R}$ be a simple ring. Then: every morphism ${\displaystyle R\to R}$ is either zero or an isomorphism. (Schur's lemma)

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