Mathematics 894 Problem Set 6

Questions due after reading week (Friday February 29).

1. Let R be a commutative ring with 1, and I, J ideals of R.

(a) Prove that if N is a submodule of the R-module M then I(M/N) ≅ (N + IM)/N.

(b) Use part (a) and the fact that for any R-module M we have M ⊗_{R} R/I ≅ M/IM to show that R/I ⊗_{R} R/J ≅ R/(I + J). State explicitly what this isomorphism does to simple tensors.

(c) What can we conclude about ℤ/nℤ ⊗_{ℤ} ℤ/mℤ?

2. Let p be a prime number and let ℤ_{p^{∞}} be the subgroup of ℚ/ℤ consisting of the canonical images in ℚ/ℤ of all a/p^{n} such that a ∈ ℤ, n ∈ ℕ. Prove that ℤ_{p^{∞}} is divisible.

3. Page 435, #2.

4. Page 455, #8. Note: We have finished discussing sections 10.4, 10.5. Then we discussed dual modules, which work over an arbitrary ring (with 1, but not necessarily commutative), not just fields, as is done in section 11.3. Then we discussed tensor, symmetric and exterior algebras from section 11.5. These work only over a commutative ring R.