Module examples

Question

I'm looking for some simple counterexamples to prove that the following statements are false:

Let R be a commutative ring with 1 and M an R-module

a) M free ⇒ M indecomposable

b) M cyclic ⇒ M indecomposable

c) M cyclic ⇒ M free

d) M indecomposable ⇒ M cyclic

For a): I thought of ℤ² since ℤ² = ℤxℤ is free but ℤ² can be decomposed into ℤ² ≅ ℤ ⊕ ℤ

For b): I thought of ℤ/6ℤ as it is cyclic (because it is generated by the unity element) but decomposable into ℤ/6ℤ ≅ ℤ/3ℤ ⊕ ℤ/2ℤ

For c): maybe ℤ/2ℤ?

I can't think of a counterexample for c) and d) via ℤ. I would be very grateful if someone would help me!

Answer 1

c) ℤ/2ℤ is cyclic but not free (it has torsion).

d) $(\mathbb{Q},+)$ is indecomposable ( Why is the additive group of rational numbers indecomposable?) but not cyclic.