Answer to Question #16723 in Abstract Algebra for john.george.milnor

Question

Let R be a domain. If R has a minimal left ideal, show that R is a division ring.

Answer 1

Let I ⊆ R be a minimal left ideal, and fix an element a _= 0 in I. Then I = Ra = Ra^2. In particular, a = ra^2 for some r ∈ R. Cancelling a, we have 1 = ra ∈ I, so I = R. The minimality of I shows that R has no left ideals other than (0) and R, so R is a division ring.