Topics in Abstract Algebra/Non-commutative rings

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

{{Template:BOOKTEMPLATE/Theorem |theorem=Proposition |label= |claim=Let $R$ be a simple ring. Then: every morphism $R\to R$ is either zero or an isomorphism. (Schur's lemma)}}

{{Template:BOOKTEMPLATE/Theorem |theorem=Theorem |label= |name=Levitzky |claim=Let $R$ be a right noetherian ring. Then every (left or right) nil ideal is nilpotent.}}