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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t0933301.png" /> be an associative ring with unit, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t0933302.png" /> a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t0933303.png" />-module. The torsion subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t0933304.png" /> is defined as
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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/t/t093/t093330/t0933305.png" /></td> </tr></table>
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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/t/t093/t093330/t0933306.png" /></td> </tr></table>
+
Let  $  R $
 +
be an associative ring with unit, and  $  M $
 +
a left  $  R $-module. The torsion subgroup  $  T( M) $
 +
is defined as
  
Here a regular element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t0933307.png" /> is an element that is not a zero divisor (neither left nor right).
+
$$
 +
T( M) =
 +
\{ {x \in M } : {
 +
\operatorname{Ann} _ {R} ( x) \textrm{ contains  a  regular  element  } }
 +
\} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t0933308.png" /> is left Ore (cf. below and [[Associative rings and algebras|Associative rings and algebras]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t0933309.png" /> is a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333010.png" />, called the torsion submodule. A module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333011.png" /> is torsion free if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333012.png" />. A module is torsion if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333013.png" />.
+
Here a regular element  $  r \in R $
 +
is an element that is not a zero divisor (neither left nor right).
  
Quite generally, a torsion theory for an Abelian category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333014.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333015.png" /> of subclasses of the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333021.png" /> are maximal with this property, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333024.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333027.png" />.
+
If  $  R $
 +
is left Ore (cf. below and [[Associative rings and algebras|Associative rings and algebras]]), then  $  T( M) $
 +
is a submodule of $  M $,
 +
called the torsion submodule. A module  $  M $
 +
is torsion free if $  T( M) = \{ 0 \} $.  
 +
A module is torsion if $  T( M) = M $.
  
The torsion submodules and torsion-free submodules of a left Ore ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333028.png" /> from a torsion theory for the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333029.png" /> of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333030.png" />-modules.
+
Quite generally, a torsion theory for an Abelian category  $  {\mathcal C} $
 +
is a pair  $  ( {\mathcal T} , {\mathcal F} ) $
 +
of subclasses of the objects of  $  {\mathcal C} $
 +
such that  $  \operatorname{Mor} _  {\mathcal C}  ( T, F  ) = \{ 0 \} $
 +
for all  $  T \in {\mathcal T} $,
 +
$  F \in {\mathcal F} $
 +
and  $  {\mathcal T} $
 +
and  $  {\mathcal F} $
 +
are maximal with this property, i.e. if  $  \operatorname{Mor} _  {\mathcal C}  ( X, F  ) = \{ 0 \} $
 +
for all  $  F \in {\mathcal F} $,
 +
then  $  X \in {\mathcal T} $,
 +
and if  $  \operatorname{Mor} _  {\mathcal C}  ( T, Y) = \{ 0 \} $
 +
for all  $  T \in {\mathcal T} $,
 +
then  $  Y \in {\mathcal F} $.
  
A radical on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333031.png" /> is a left-exact functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333032.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333033.png" />,
+
The torsion submodules and torsion-free submodules of a left Ore ring  $  R $
 +
from a torsion theory for the category  $  R \textrm{ - Mod  } $
 +
of left $  R $-modules.
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333034.png" /> is a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333035.png" />;
+
A radical on  $  R \textrm{ - Mod  } $
 +
is a left-exact functor  $  \operatorname{Rad} : R \textrm{ - Mod  } \rightarrow R \textrm{ - Mod  } $
 +
such that for all  $  M, N \in R \textrm{ - Mod  } $,
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333037.png" />; more precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333038.png" /> is the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333039.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333040.png" />.
+
i) $  \operatorname{Rad} ( M) $
 +
is a submodule of $  M $;
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333041.png" />.
+
ii) $  f ( \operatorname{Rad}  M ) \subset  \operatorname{Rad}  N $
 +
for all  $  f \in \Hom {R} ( M, N) $;
 +
more precisely,  $  \operatorname{Rad} ( f  ) $
 +
is the restriction of  $  f $
 +
to  $  \operatorname{Rad} ( M) \subset  M $.
  
A radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333042.png" /> is a torsion radical or hereditary radical if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333043.png" /> for each submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333044.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333045.png" />. A torsion radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333046.png" /> defines a torsion theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333047.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333049.png" />. All torsion theories for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333050.png" /> arise this way.
+
iii)  $  \operatorname{Rad} ( M / \operatorname{Rad} ( M) ) = \{ 0 \} $.
  
A left denominator set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333051.png" /> is a submonoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333053.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333055.png" />) such that:
+
A radical  $  \operatorname{Rad} $
 +
is a torsion radical or hereditary radical if  $  N \cap \operatorname{Rad} ( M) = \operatorname{Rad} ( N) $
 +
for each submodule  $  N $
 +
of a module  $  M $.  
 +
A torsion radical  $  \operatorname{Rad} $
 +
defines a torsion theory for  $  R \textrm{ - Mod  } $
 +
with  $  {\mathcal T} _ {\operatorname{\rm Rad}  } = \{ {M \in R \textrm{ - Mod  } } : { \operatorname{Rad} ( M) = M } \} $,
 +
$  {\mathcal F} _ { \operatorname{Rad}  } = \{ {M \in R \textrm{ - Mod  } } : { \operatorname{Rad} ( M) = 0 } \} $.  
 +
All torsion theories for  $  R \textrm{ - Mod  } $
 +
arise this way.
  
a) (the left Ore condition) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333057.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333060.png" />;
+
A left denominator set of  $  R $
 +
is a submonoid  $  S $
 +
of  $  R $ (i.e. $  1 \in S $
 +
and  $  s _ {1} , s _ {2} \in S \Rightarrow s _ { 1 _ 2 } \in S $)
 +
such that:
  
b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333061.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333063.png" />, then there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333064.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333065.png" />.
+
a) (the left Ore condition) for all  $  s _ {1} \in S $,  
 +
$  r _ {1} \in R $
 +
there are  $  s _ {2} \in S $,  
 +
$  r _ {2} \in R $
 +
such that  $  s _ {2} r _ {1} = r _ {2} s _ {1} $;
  
If the set of all regular elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333066.png" /> is a left denominator set, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333067.png" /> is called left Ore. A left denominator set is also called a left Ore set.
+
b) if  $  r s = 0 $
 +
for  $  r \in R $,
 +
$  s \in S $,  
 +
then there is an  $  s  ^  \prime  \in S $
 +
with  $  s  ^  \prime  r = 0 $.
  
A left denominator set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333068.png" /> defines a torsion theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333069.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093330/t09333070.png" /> by the associated radical functor
+
If the set of all regular elements of  $  R $
 +
is a left denominator set, then  $  R $
 +
is called left Ore. A left denominator set is also called a left Ore set.
  
<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/t/t093/t093330/t09333071.png" /></td> </tr></table>
+
A left denominator set  $  S $
 +
defines a torsion theory  $  ( {\mathcal T} _ {s} , {\mathcal F} _ {s} ) $
 +
for  $  R \textrm{ - Mod  } $
 +
by the associated radical functor
 +
 
 +
$$
 +
\operatorname{Rad} _ {S} ( M )  = \
 +
\{ {x \in M } : { \operatorname{Ann} _ {R} ( x) \cap S \neq \emptyset } \}
 +
.
 +
$$
  
 
This illustrates the links between torsion theories and (non-commutative) localization (theories). For much more about this theme cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], cf. also [[Fractions, ring of|Fractions, ring of]].
 
This illustrates the links between torsion theories and (non-commutative) localization (theories). For much more about this theme cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], cf. also [[Fractions, ring of|Fractions, ring of]].

Latest revision as of 07:14, 12 July 2022


Let $ R $ be an associative ring with unit, and $ M $ a left $ R $-module. The torsion subgroup $ T( M) $ is defined as

$$ T( M) = \{ {x \in M } : { \operatorname{Ann} _ {R} ( x) \textrm{ contains a regular element } } \} . $$

Here a regular element $ r \in R $ is an element that is not a zero divisor (neither left nor right).

If $ R $ is left Ore (cf. below and Associative rings and algebras), then $ T( M) $ is a submodule of $ M $, called the torsion submodule. A module $ M $ is torsion free if $ T( M) = \{ 0 \} $. A module is torsion if $ T( M) = M $.

Quite generally, a torsion theory for an Abelian category $ {\mathcal C} $ is a pair $ ( {\mathcal T} , {\mathcal F} ) $ of subclasses of the objects of $ {\mathcal C} $ such that $ \operatorname{Mor} _ {\mathcal C} ( T, F ) = \{ 0 \} $ for all $ T \in {\mathcal T} $, $ F \in {\mathcal F} $ and $ {\mathcal T} $ and $ {\mathcal F} $ are maximal with this property, i.e. if $ \operatorname{Mor} _ {\mathcal C} ( X, F ) = \{ 0 \} $ for all $ F \in {\mathcal F} $, then $ X \in {\mathcal T} $, and if $ \operatorname{Mor} _ {\mathcal C} ( T, Y) = \{ 0 \} $ for all $ T \in {\mathcal T} $, then $ Y \in {\mathcal F} $.

The torsion submodules and torsion-free submodules of a left Ore ring $ R $ from a torsion theory for the category $ R \textrm{ - Mod } $ of left $ R $-modules.

A radical on $ R \textrm{ - Mod } $ is a left-exact functor $ \operatorname{Rad} : R \textrm{ - Mod } \rightarrow R \textrm{ - Mod } $ such that for all $ M, N \in R \textrm{ - Mod } $,

i) $ \operatorname{Rad} ( M) $ is a submodule of $ M $;

ii) $ f ( \operatorname{Rad} M ) \subset \operatorname{Rad} N $ for all $ f \in \Hom {R} ( M, N) $; more precisely, $ \operatorname{Rad} ( f ) $ is the restriction of $ f $ to $ \operatorname{Rad} ( M) \subset M $.

iii) $ \operatorname{Rad} ( M / \operatorname{Rad} ( M) ) = \{ 0 \} $.

A radical $ \operatorname{Rad} $ is a torsion radical or hereditary radical if $ N \cap \operatorname{Rad} ( M) = \operatorname{Rad} ( N) $ for each submodule $ N $ of a module $ M $. A torsion radical $ \operatorname{Rad} $ defines a torsion theory for $ R \textrm{ - Mod } $ with $ {\mathcal T} _ {\operatorname{\rm Rad} } = \{ {M \in R \textrm{ - Mod } } : { \operatorname{Rad} ( M) = M } \} $, $ {\mathcal F} _ { \operatorname{Rad} } = \{ {M \in R \textrm{ - Mod } } : { \operatorname{Rad} ( M) = 0 } \} $. All torsion theories for $ R \textrm{ - Mod } $ arise this way.

A left denominator set of $ R $ is a submonoid $ S $ of $ R $ (i.e. $ 1 \in S $ and $ s _ {1} , s _ {2} \in S \Rightarrow s _ { 1 _ 2 } \in S $) such that:

a) (the left Ore condition) for all $ s _ {1} \in S $, $ r _ {1} \in R $ there are $ s _ {2} \in S $, $ r _ {2} \in R $ such that $ s _ {2} r _ {1} = r _ {2} s _ {1} $;

b) if $ r s = 0 $ for $ r \in R $, $ s \in S $, then there is an $ s ^ \prime \in S $ with $ s ^ \prime r = 0 $.

If the set of all regular elements of $ R $ is a left denominator set, then $ R $ is called left Ore. A left denominator set is also called a left Ore set.

A left denominator set $ S $ defines a torsion theory $ ( {\mathcal T} _ {s} , {\mathcal F} _ {s} ) $ for $ R \textrm{ - Mod } $ by the associated radical functor

$$ \operatorname{Rad} _ {S} ( M ) = \ \{ {x \in M } : { \operatorname{Ann} _ {R} ( x) \cap S \neq \emptyset } \} . $$

This illustrates the links between torsion theories and (non-commutative) localization (theories). For much more about this theme cf. [a1], [a2], [a3], cf. also Fractions, ring of.

References

[a1] L.H. Rowen, "Ring theory" , 1 , Acad. Press (1988) pp. §3.4
[a2] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) pp. §15, §16
[a3] J.S. Golan, "Localization of noncommutative rings" , M. Dekker (1975)
How to Cite This Entry:
Torsion submodule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_submodule&oldid=18242