In Constructive Mathematics
Flat modules have increased importance in constructive mathematics, where projective modules are less useful. For example, that all free modules are projective is equivalent to the full axiom of choice, so theorems about projective modules, even if proved constructively, do not necessarily apply to free modules. In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply.
Read more about this topic: Flat Module
Famous quotes containing the words constructive and/or mathematics:
“If grandparents want to have a meaningful and constructive role, the first lesson they must learn is that becoming a grandparent is not having a second chance at parenthood!”
—Eda Le Shan (20th century)
“I must study politics and war that my sons may have liberty to study mathematics and philosophy.”
—John Adams (1735–1826)