Math Help - Direct sum of free modules

1. Direct sum of free modules

I encountered this sentence, but I could not understand it.

The direct sum of free modules is free. Take the union of the basis elements over each component to build a basis for M. But the converse is not true. Z6 is the direct sum of Z2 and Z3, yet the two summands are not free Z6 modules.

Zn (Z subscript n) denotes the set of integers modulo n.

What are the examples of free Z6 modules?
Why Z2 and Z3 cannot be free Z6 modules?

Thanks in advance.

2. Originally Posted by vagabond

What are the examples of free Z6 modules?
$\mathbb{Z}_6, \ \mathbb{Z}_6 \oplus \mathbb{Z}_6,$ etc.

Why Z2 and Z3 cannot be free Z6 modules?
because they are not faithful $\mathbb{Z}_6$ modules: $\mathbb{Z}_2$ is annihilated by $\bar{2} \in \mathbb{Z}_6$ and $\mathbb{Z}_3$ is annihilated by $\bar{3} \in \mathbb{Z}_6.$