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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|Projective module]]; [[Free 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|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|Adic topology]]). The classical ring of fractions of a ring  $  R $
is a flat module over  $  R $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Lambek,   "Lectures on rings and modules" , Blaisdell  (1966)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Commutative algebra" , Addison-Wesley  (1964)  (Translated from French)</TD></TR></table>