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 .
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
|[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)|
Torsion submodule. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Torsion_submodule&oldid=18242