Show that rad R is the intersection of all modular maximal left (resp. right) ideals of R.
1
Expert's answer
2012-10-30T10:32:35-0400
Let J be the intersection of all modular maximal left ideals of R. (If there are no modular maximal left ideals, we define J = R.) First we prove J ⊆rad R. Consider any a not fromrad R. Then e: = xa is not left quasi-regular for some x ∈ R. It is easy to check that I = {r −re : r ∈ R} is a modular left ideal. Since e is not in I, we have I is less than R. I ⊆m for some modular maximal left ideal m, with e not fromm.In particular, e = xa is not in J, so a is not in J.We now finish by proving that rad R ⊆ J. Assume the contrary. Then rad R is not contained in m for some modular maximal left ideal m. In particular, rad R +m = R. Let e ∈ R be such that r ≡ re (mod m) for all r ∈ R. Write e = a + b where a ∈rad R and b ∈m. Then e − a ∈m so e − ae ∈m. Since rad R is left quasi-regular, there exists a' ∈ R such that a' + a − a'a = 0. But then e = e − (a' + a − a'a)e =(e − ae) − a_(e − ae) ∈m, a contradiction.
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