189-571B: Higher Algebra II
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...
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'=EndR(M) and R''=EndR'(M). ).
c) Conclude that the ring Mn(F) of n by n matrices with entries in a field F (or, more generally, in a division ring)
Recall from class that Mn(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,..., xt and arbitrary vectors
there exists a transformation a in S such that
a xj = yj, for j=1,...,t.
Show that a ring of linear
transformations that is two-fold transitive is