189-571B: Higher Algebra II
There will be no problem
session on the week after the study break, since this hour
will be taken up with the mid-term exam.
The following questions make up a "practice mid-term"
and need not be handed in.
1. Describe completely all the maximal and minimal right
ideals of the ring R=
EndF(V), where V is a finite-dimensional
vector space over a field F. Refrain from choosing bases for V and from
identifying R with Mn(F).
2. Show that the group algebra F[G] is semi-simple if the characteristic of F
does not divide the order of G.
Give an example where F[G] is not semi-simple.
3. Give an example of a module V over an algebra R for which the
conclusion of the density theorem fails, i.e., for which
R is not dense in R'' (in the notations that we used in
stating the theorem).
4. Show that the only division algebras over R are R, C, and H.
5. Show that there are no finite non-commutative division algebras.
6. Give an example of a non-commutative division algebra
over Q, the field of rational numbers.