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Topics in Abstract Algebra/Non-commutative rings

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.

Category:Book:Topics in Abstract Algebra#Non-commutative%20rings%20

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