# 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) |

#### Comments

#### References

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

**How to Cite This Entry:**

Flat module.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Flat_module&oldid=17725