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:
“Friendship among nations, as among individuals, calls for constructive efforts to muster the forces of humanity in order that an atmosphere of close understanding and cooperation may be cultivated.”
—Franklin D. Roosevelt (1882–1945)
“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o’ th’ day
The clock doth strike, by algebra.”
—Samuel Butler (1612–1680)