62 494
Assignments Done
Successfully Done
In June 2018

Answer to Question #23263 in Abstract Algebra for Melvin Henriksen

Question #23263
Let R = End(Vk) where V is a vector space over a division ring k. Show that all ideals of R are linearly ordered by inclusion and idempotent.
Expert's answer
If dimk V <∞, R isa simple ring.
Therefore, it suffices to treat thecase when V is infinite-dimensional.
The ideals of R are linearlyordered by inclusion.
To show that they are allidempotent, consider any ideal nonzero I.
There exists an infinite cardinal β <dimk V such that
I = {f R : dimk f(V ) < β}.
For any f I, let f' R be such that f' is theidentity on the f(V ), and zero on a direct complement of f(V). Clearly, f' I, and f= f'f. Therefore, f I^2, and wehave proved that I = I^2.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!


No comments. Be first!

Leave a comment

Ask Your question

Privacy policy Terms and Conditions