**Yoshifumi Tsuchimoto**

- is a direct summand of free modules.
- is projective

An
-module
is said to be **divisible** if
for any
, the multplication map

is surjective.

An
-module
is said to be **divisible** if
for any
, the multplication map

is epic.

For the proof of the proposition above, we need the followin lemmas.

- For any free -module , is divisible (hence is -injective).
- For any -module , there is a canonical injective -homomorphism .
- Any -module may be embeded in a divisible module .