# 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: *
every morphism 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.*