Torsion submodule

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)