# 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.