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

Question #16723

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

Expert's answer

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.
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