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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