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:
“Euphemisms are not, as many young people think, useless verbiage for that which can and should be said bluntly; they are like secret agents on a delicate mission, they must airily pass by a stinking mess with barely so much as a nod of the head, make their point of constructive criticism and continue on in calm forbearance. Euphemisms are unpleasant truths wearing diplomatic cologne.”
—Quentin Crisp (b. 1908)
“Why does man freeze to death trying to reach the North Pole? Why does man drive himself to suffer the steam and heat of the Amazon? Why does he stagger his mind with the mathematics of the sky? Once the question mark has arisen in the human brain the answer must be found, if it takes a hundred years. A thousand years.”
—Walter Reisch (1903–1963)