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