 
Summary: Rings and Algebras Problem set #3. Sept. 29, 2011.
1. Find a semiprimitive ring R which has a unique nontrivial twosided ideal.
2. Let V be a (not necessarily finite dimensional) vector space. Find the Jacobson radical of the
exterior algebra #(V ).
3. Prove that if R is a principal ideal domain with infinitely many prime elements then J(R) = 0.
4. A submodule N # M is called superfluous in M if for any submodule K # M we have that if
K +N = M then K = M . Prove that J(R) is always a superfluous left ideal in R but give an
example of a module M where rad M is not superfluous. (Recall that the radical of a module
is the intersection of its maximal submodules.)
5. What is the Jacobson radical of an ultraproduct of the rings R i , i # I?
6. Show that if e = e 2
# J(R) then e = 0.
7. Recall that the socle of a module M , denoted by soc M is the sum of its simple submodules.
Show that soc M # {m # M  J(R)m = 0} and when R/J(R) is left artinian, there is an
equality. Give an example when the equality does not hold.
8. A ring R is von Neumann regular if for every r # R there is s # R such that rsr = r.
a) Show that End(VD ) is always von Neumann regular.
b) Show that if R is von Neumann regular then J(R) = 0 (i. e. R is semiprimitive).
9. Let X be an arbitrary set R X be the ring of all real valued maps on X with pointwise operations.
When X is endowed with a topology, let C(X) denote the subring of continuous functions.
