Basic Ring Theory - 23 (Absolute Flatness)

[Tsakanikas Nickos]

A ring $ A $ is called absolutely flat if every $ A $-module is flat.

Show that for a ring $ A $, the following are equivalent:

1. $ A $ is absolutely flat.
2. Every principal ideal is idempotent.
3. Every finitely generated ideal is a direct summand of $ A $.

Moreover, show that if a local ring is absolutely flat, then it is a field.