Rings and Algebras Problem set #3. Sept. 29, 2011. 1. Find a semiprimitive ring R which has a unique non-trivial two-sided ideal. Summary: Rings and Algebras Problem set #3. Sept. 29, 2011. 1. Find a semiprimitive ring R which has a unique non-trivial two-sided 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 Ri, i I? 6. Show that if e = e2 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 RX be the ring of all real valued maps on X with pointwise operations. Collections: Mathematics