# Torsion submodule

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Let be an associative ring with unit, and a left -module. The torsion subgroup is defined as  Here a regular element is an element that is not a zero divisor (neither left nor right).

If is left Ore (cf. below and Associative rings and algebras), then is a submodule of , called the torsion submodule. A module is torsion free if . A module is torsion if .

Quite generally, a torsion theory for an Abelian category is a pair of subclasses of the objects of such that for all , and and are maximal with this property, i.e. if for all , then , and if for all , then .

The torsion submodules and torsion-free submodules of a left Ore ring from a torsion theory for the category of left -modules.

A radical on is a left-exact functor such that for all ,

i) is a submodule of ;

ii) for all ; more precisely, is the restriction of to .

iii) .

A radical is a torsion radical or hereditary radical if for each submodule of a module . A torsion radical defines a torsion theory for with , . All torsion theories for arise this way.

A left denominator set of is a submonoid of (i.e. and ) such that:

a) (the left Ore condition) for all , there are , such that ;

b) if for , , then there is an with .

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

A left denominator set defines a torsion theory for by the associated radical functor 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.