Let R be a domain. If R has a minimal left ideal, show that R is a division ring.
1
Expert's answer
2012-10-22T11:16:06-0400
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.
Numbers and figures are an essential part of our world, necessary for almost everything we do every day. As important…
APPROVED BY CLIENTS
"assignmentexpert.com" is professional group of people in Math subjects! They did assignments in very high level of mathematical modelling in the best quality. Thanks a lot
Comments
Leave a comment