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Difference between revisions of "Torsion submodule"

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m (fixing spaces)
Line 13: Line 13:
 
Let  $  R $
 
Let  $  R $
 
be an associative ring with unit, and  $  M $
 
be an associative ring with unit, and  $  M $
a left  $  R $-
+
a left  $  R $-module. The torsion subgroup  $  T( M) $
module. The torsion subgroup  $  T( M) $
 
 
is defined as
 
is defined as
  
 
$$  
 
$$  
 
T( M) =
 
T( M) =
$$
 
 
$$
 
= \
 
 
\{ {x \in M } : {
 
\{ {x \in M } : {
  \mathop{\rm Ann} _ {R} ( x)  \textrm{ contains  a  regular  element  } }
+
  \operatorname{Ann} _ {R} ( x)  \textrm{ contains  a  regular  element  } }
 
\} .
 
\} .
 
$$
 
$$
Line 41: Line 36:
 
is a pair  $  ( {\mathcal T} , {\mathcal F} ) $
 
is a pair  $  ( {\mathcal T} , {\mathcal F} ) $
 
of subclasses of the objects of  $  {\mathcal C} $
 
of subclasses of the objects of  $  {\mathcal C} $
such that  $  \mathop{\rm Mor} _  {\mathcal C}  ( T, F  ) = \{ 0 \} $
+
such that  $  \operatorname{Mor} _  {\mathcal C}  ( T, F  ) = \{ 0 \} $
 
for all  $  T \in {\mathcal T} $,  
 
for all  $  T \in {\mathcal T} $,  
 
$  F \in {\mathcal F} $
 
$  F \in {\mathcal F} $
 
and  $  {\mathcal T} $
 
and  $  {\mathcal T} $
 
and  $  {\mathcal F} $
 
and  $  {\mathcal F} $
are maximal with this property, i.e. if  $   \mathop{\rm Mor} _  {\mathcal C}  ( X, F  ) = \{ 0 \} $
+
are maximal with this property, i.e. if  $ \operatorname{Mor} _  {\mathcal C}  ( X, F  ) = \{ 0 \} $
 
for all  $  F \in {\mathcal F} $,  
 
for all  $  F \in {\mathcal F} $,  
 
then  $  X \in {\mathcal T} $,  
 
then  $  X \in {\mathcal T} $,  
and if  $  \mathop{\rm Mor} _  {\mathcal C}  ( T, Y) = \{ 0 \} $
+
and if  $  \operatorname{Mor} _  {\mathcal C}  ( T, Y) = \{ 0 \} $
 
for all  $  T \in {\mathcal T} $,  
 
for all  $  T \in {\mathcal T} $,  
 
then  $  Y \in {\mathcal F} $.
 
then  $  Y \in {\mathcal F} $.
Line 55: Line 50:
 
The torsion submodules and torsion-free submodules of a left Ore ring  $  R $
 
The torsion submodules and torsion-free submodules of a left Ore ring  $  R $
 
from a torsion theory for the category  $  R \textrm{ - Mod  } $
 
from a torsion theory for the category  $  R \textrm{ - Mod  } $
of left  $  R $-
+
of left  $  R $-modules.
modules.
 
  
 
A radical on  $  R \textrm{ - Mod  } $
 
A radical on  $  R \textrm{ - Mod  } $
is a left-exact functor  $   \mathop{\rm Rad} :  R \textrm{ - Mod  } \rightarrow 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  } $,
 
such that for all  $  M, N \in R \textrm{ - Mod  } $,
  
i)  $   \mathop{\rm Rad} ( M) $
+
i)  $ \operatorname{Rad} ( M) $
 
is a submodule of  $  M $;
 
is a submodule of  $  M $;
  
ii)  $  f ( \mathop{\rm Rad}  M ) \subset  \mathop{\rm Rad}  N $
+
ii)  $  f ( \operatorname{Rad}  M ) \subset  \operatorname{Rad}  N $
for all  $  f \in \mathop{\rm Hom} _ {R} ( M, N) $;  
+
for all  $  f \in \Hom} {R} ( M, N) $;  
more precisely,  $   \mathop{\rm Rad} ( f  ) $
+
more precisely,  $ \operatorname{Rad} ( f  ) $
 
is the restriction of  $  f $
 
is the restriction of  $  f $
to  $   \mathop{\rm Rad} ( M) \subset  M $.
+
to  $ \operatorname{Rad} ( M) \subset  M $.
  
iii)  $   \mathop{\rm Rad} ( M / \mathop{\rm Rad} ( M) ) = \{ 0 \} $.
+
iii)  $ \operatorname{Rad} ( M / \operatorname{Rad} ( M) ) = \{ 0 \} $.
  
A radical  $   \mathop{\rm Rad} $
+
A radical  $ \operatorname{Rad} $
is a torsion radical or hereditary radical if  $  N \cap \mathop{\rm Rad} ( M) =  \mathop{\rm Rad} ( N) $
+
is a torsion radical or hereditary radical if  $  N \cap \operatorname{Rad} ( M) =  \operatorname{Rad} ( N) $
 
for each submodule  $  N $
 
for each submodule  $  N $
 
of a module  $  M $.  
 
of a module  $  M $.  
A torsion radical  $   \mathop{\rm Rad} $
+
A torsion radical  $ \operatorname{Rad} $
 
defines a torsion theory for  $  R \textrm{ - Mod  } $
 
defines a torsion theory for  $  R \textrm{ - Mod  } $
with  $  {\mathcal T} _ { \mathop{\rm Rad}  } = \{ {M \in R \textrm{ - Mod  } } : { \mathop{\rm Rad} ( M) = M } \} $,  
+
with  $  {\mathcal T} _ {\operatorname{\rm Rad}  } = \{ {M \in R \textrm{ - Mod  } } : { \operatorname{Rad} ( M) = M } \} $,  
$  {\mathcal F} _ { \mathop{\rm Rad}  } = \{ {M \in R \textrm{ - Mod  } } : { \mathop{\rm Rad} ( M) = 0 } \} $.  
+
$  {\mathcal F} _ { \operatorname{Rad}  } = \{ {M \in R \textrm{ - Mod  } } : { \operatorname{Rad} ( M) = 0 } \} $.  
 
All torsion theories for  $  R \textrm{ - Mod  } $
 
All torsion theories for  $  R \textrm{ - Mod  } $
 
arise this way.
 
arise this way.
Line 86: Line 80:
 
A left denominator set of  $  R $
 
A left denominator set of  $  R $
 
is a submonoid  $  S $
 
is a submonoid  $  S $
of  $  R $(
+
of  $  R $ (i.e.  $  1 \in S $
i.e.  $  1 \in S $
 
 
and  $  s _ {1} , s _ {2} \in S \Rightarrow s _ { 1 _ 2 } \in S $)  
 
and  $  s _ {1} , s _ {2} \in S \Rightarrow s _ { 1 _ 2 } \in S $)  
 
such that:
 
such that:
Line 113: Line 106:
  
 
$$  
 
$$  
  \mathop{\rm Rad} _ {S} ( M )  = \  
+
  \operatorname{Rad} _ {S} ( M )  = \  
\{ {x \in M } : { \mathop{\rm Ann} _ {R} ( x) \cap S \neq \emptyset } \}
+
\{ {x \in M } : { \operatorname{Ann} _ {R} ( x) \cap S \neq \emptyset } \}
 
.
 
.
 
$$
 
$$

Revision as of 07:12, 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=48999