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A left (or right) module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405901.png" /> over an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405902.png" /> such that the tensor-product functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405903.png" /> (correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405904.png" />) is exact. This definition is equivalent to any of the following: 1) the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405905.png" /> (correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405906.png" />); 2) the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405907.png" /> can be represented in the form of a direct (injective) limit of summands of free modules; 3) the character module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405908.png" /> is injective, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f0405909.png" /> is the group of rational numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059010.png" /> is the group of integers; and 4) for any right (correspondingly, left) ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059012.png" />, the canonical homomorphism
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059013.png" /></td> </tr></table>
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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059014.png" /> are flat modules if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059015.png" /> is regular in the sense of von Neumann (see [[Absolutely-flat ring|Absolutely-flat ring]]). A coherent ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059016.png" /> can be defined as a ring over which the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059017.png" /> of any number of copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059018.png" /> is a flat module. The operations of localization and completion with respect to powers of an ideal of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059019.png" /> lead to flat modules over the ring (see [[Adic topology|Adic topology]]). The classical ring of fractions of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059020.png" /> is a flat module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040590/f04059021.png" />.
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Commutative algebra" , Addison-Wesley  (1964)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Commutative algebra" , Addison-Wesley  (1964)  (Translated from French)</TD></TR></table>

Latest revision as of 19:39, 5 June 2020


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=46941
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article