Rings and Algebras Problem set #2. Sept. 22, 2010. 1. Show that the converse of Schur's lemma does not hold. Summary: Rings and Algebras Problem set #2. Sept. 22, 2010. 1. Show that the converse of Schur's lemma does not hold. 2. Let V be a vector space over a field K. Show that R = End(VK) is left primitive but not necessarily simple. Describe the ideal structure of R. 3. Show that if R = End(VK) as above then R has a minimal left ideal and conclude that every simple faithful left R-module is isomorphic to RV . 4. Let R be a left primitive ring and 0 = e R an idempotent element. Show that S = eRe is also left primitive. 5. Show that for K a field of characteristic 0 the Weyl-algebra A1(K) = K x, y /(xy - yx - 1) is left primitive. 6. Decide whether the following implications are true: a) A ring R is left primitive if and only if the full matrix ring Mn(R) is left primitve. b) A ring R is prime if and only if the full matrix ring Mn(R) is prime. 7. a) Suppose the path algebra KG is finite dimensional. Give a precise condition for KG to be primitive (prime, resp.). b) Show that KG (without the assumption on the dimension) is prime if and only if for each pair of vertices i, j in G there is an (oriented) path from i to j. 8. Show that the fact that R End(VD) is 1-transitive does not imply that R is dense in End(VD) (although it follows that V is a simple faithful R-module). (Hint: Construct an example where End(RV ) is strictly larger than D.) Collections: Mathematics