Torsion submodule
Let be an associative ring with unit, and
a left
-module. The torsion subgroup
is defined as
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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
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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) |
Torsion submodule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_submodule&oldid=48999