Flat module

A left (or right) module $ M $ over an associative ring $ R $ such that the tensor-product functor $ \otimes _ {R} M $( correspondingly, $ M \otimes _ {R} $) is exact. This definition is equivalent to any of the following: 1) the functor $ \mathop{\rm Tor} _ {1} ^ {R} (-, M) = 0 $( correspondingly, $ \mathop{\rm Tor} _ {1} ^ {R} ( M, -) = 0 $); 2) the module $ M $ can be represented in the form of a direct (injective) limit of summands of free modules; 3) the character module $ M ^ {*} = \mathop{\rm Hom} _ {\mathbf Z } ( M, \mathbf Q / \mathbf Z ) $ is injective, where $ \mathbf Q $ is the group of rational numbers and $ \mathbf Z $ is the group of integers; and 4) for any right (correspondingly, left) ideal $ J $ of $ R $, the canonical homomorphism

$$ J \otimes _ {R} M \rightarrow JM \ \ ( M\otimes _ {R} J \rightarrow MJ) $$

is an isomorphism.

Projective modules and free modules are examples of flat modules (cf. Projective module; Free module). The class of flat modules over the ring of integers coincides with the class of Abelian groups without torsion. All modules over a ring $ R $ are flat modules if and only if $ R $ is regular in the sense of von Neumann (see Absolutely-flat ring). A coherent ring $ R $ can be defined as a ring over which the direct product $ \prod R _ \alpha $ of any number of copies of $ R $ is a flat module. The operations of localization and completion with respect to powers of an ideal of a ring $ R $ lead to flat modules over the ring (see Adic topology). The classical ring of fractions of a ring $ R $ is a flat module over $ R $.

References

[1]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2]	J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)

References

[a1]	N. Bourbaki, "Commutative algebra" , Addison-Wesley (1964) (Translated from French)