Topics in Abstract Algebra/Non-commutative rings

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

Proposition. Let R be a simple ring. Then

  • (i) Every morphism R \to R is either zero or an isomorphism. (Schur's lemma)
  • (ii)

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