Rings and Algebras Problem set #3. Sept. 29, 2011. 1. Find a semiprimitive ring R which has a unique nontrivial twosided ideal. 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. Collections: Mathematics