189-571B: Higher Algebra II

Assignment 4

To be presented: Wednesday, February 7.

The problem marked with a (*) is for everyone to do and to hand in on the day of the problem session. I will return this problem to you on Friday of the same week, so late problems will not be accepted!

If you are stuck on your problem, come see me during my Monday office hours (or make an appointment) to get a hint...

1. (*) Let R be a primitive ring and let M be a faithful irreducible R-module. If I is an ideal of R, let M[I] denote the I-torsion submodule of M, i.e., the set of m in M such that am=0 for all a in I.

a) Show that IM=M, that M[I]=0, and that Ix=Rx for all x in M.

b) Modify the proof of the density theorem to show that I is dense in R'' (with notations as in the statement of the theorem given in class, so that R'=End_R(M) and R''=End_R'(M). ).

c) Conclude that the ring M_n(F) of n by n matrices with entries in a field F (or, more generally, in a division ring) is simple.

Remark: Recall from class that M_n(F) is known to be primitive, but that a primitive ring need not be simple in general.

2. (Frederic Rochon) Give an example of a completely decomposable module M over a ring R in which R'' is strictly larger than R. (Notations being as in the previous question).

3. (Dan Segal) Call a set S of linear transformations k-fold transitive if for any set of at most k linearly independent vectors x₁,..., x_t and arbitrary vectors y₁,..., y_t, there exists a transformation a in S such that a x_j = y_j, for j=1,...,t. Show that a ring of linear transformations that is two-fold transitive is dense.