Flat module

Flatness of modules is a generalization of the concept of "free module".

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. For more details see Commutative Algebra.

Definition

A module M over a ring A is called flat if the functor

Is exact. (See tensor product. )

Equivalent characterizations are:

• Tor1 (N, M) = 0 for all A -modules N. (See Tor ( mathematics). )
• For every ideal I of A

• Tor1 (A / I, M) = 0 for all ideals I of A.

Properties

• All projective and thus all the free modules are flat. Conversely, every finitely presented flat module is projective.
• Flat modules are torsion-free. About Dedekind rings (especially so over principal ideal rings ) agree the terms " flat" and " torsion " even match.
• It should be

• In the ring of dual numbers is equivalent to free flat.
• Be. Then M is flat then if all is flat.

Examples

• Is a flat, but not projective module.
• For each ring R the R- module R is flat.
• Let R be a commutative ring with unit element and a multiplicatively closed set, then the R- module is flat.