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189-571B: Higher Algebra II

Assignment 6

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.