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

$$ 
T( M) =
$$

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

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|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  $   \mathop{\rm 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  $   \mathop{\rm Mor} _  {\mathcal C}   ( X, F  ) = \{ 0 \} $
for all  $  F \in {\mathcal F} $, 
then  $  X \in {\mathcal T} $, 
and if  $   \mathop{\rm 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  $   \mathop{\rm Rad} :  R \textrm{ - Mod  } \rightarrow R \textrm{ - Mod  } $
such that for all  $  M, N \in R \textrm{ - Mod  } $,

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

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

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

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

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

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]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,   "Ring theory" , '''1''' , Acad. Press  (1988)  pp. §3.4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Faith,   "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)  pp. §15, §16</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.S. Golan,   "Localization of noncommutative rings" , M. Dekker  (1975)</TD></TR></table>