# Flat module

A left (or right) module over an associative ring such that the tensor-product functor (correspondingly, ) is exact. This definition is equivalent to any of the following: 1) the functor (correspondingly, ); 2) the module can be represented in the form of a direct (injective) limit of summands of free modules; 3) the character module is injective, where is the group of rational numbers and is the group of integers; and 4) for any right (correspondingly, left) ideal of , the canonical homomorphism

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 are flat modules if and only if is regular in the sense of von Neumann (see Absolutely-flat ring). A coherent ring can be defined as a ring over which the direct product of any number of copies of is a flat module. The operations of localization and completion with respect to powers of an ideal of a ring lead to flat modules over the ring (see Adic topology). The classical ring of fractions of a ring is a flat module over .

#### 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