MATH 611 Due Wed Oct 14, 1992

1. Prove that a necessary condition that a submodule N of an R-module M be a direct summand of M is that M/N be isomorphic to a submodule of M.

2. Let R be an integral domain, let M be an R-module and N a submodule of M. Prove that M/N is torsion free if and only if t(M) ⊆ N. (Here t(M) denotes the torsion submodule of M.)

3. Let R = ℤ and let p be a prime number. Prove that if n is not a multiple of p then the map x ↦ nx is an automorphism of ℤ(p).

4. Let R = ℤ and M = ∏_{p} ℤ(p), where the product is taken of the set of prime numbers. Recall that, by definition, ⨁_{p} ℤ(p) is a submodule of ∏_{p} ℤ(p).

a) Prove that ⨁_{p} ℤ(p) = t(M).

b) Prove that if n ≠ 0 ∈ R then the map x ↦ nx is an automorphism of M/t(M).

c) Prove that if p is a prime number and x ∈ M/t(M) then x ∈ p(M/t(M)). (In other words, there exists y ∈ M/t(M) such that x = py.)

d) Prove that if x ≠ 0 ∈ M then there exists a prime number p such that x ∉ pM.

e) Prove that t(M) is not a direct summand of M.