Download Homework #6

Math 901
Homework # 6
Due: Wednesday, December 2nd
Throughout R denotes a ring with identity.
1. Suppose A1 × · · · × Ak ∼
= B1 × · · · × B` is a ring isomorphism where Ai and Bj are
simple rings for all i and j. Prove that k = ` and (after reordering) Ai ∼
= Bi for all i.
2. P
Suppose R = R1 × · · · × Rk where each Ri is a semisimple ring. Prove that λR (R) =
∼
i λRi (Ri ). Use this to deduce that if R = Mn1 (D1 ) × · · · Mnk (Dk ) where D1 , . . . , Dk
are division rings, then λ(R) = n1 + · · · + nk .
3. Let M be a finitely generated semisimple R-module. Prove that EndR M is a semisimple ring.
In each of the problems #4-#7, let F be a field of characteristic zero and A the (first) Weyl
algebra of F .
4. Prove that A is a domain.
5. Prove that A is neither left nor right Artinian.
6. Recall that A is a subring of EndF (F [x]). Hence, F [x] is a left A-module. Prove that
F [x] is a simple A-module (and hence finitely generated).
7. Let A0 = EndA (F [x]), which is a division ring by Problem #6. Prove that F [x] is not
a finitely generated A0 -module. (Hint: A is not semisimple.)