Graded algebra
An algebra whose additive group can be represented in the form of a (weak) direct sum of groups
,
where
for any
. As a result, the additive group of a graded algebra (considered as a module over the ring of integers) is a positively graded module. As an example of a graded algebra one can quote the algebra
of polynomials over a field
, where
is the subspace generated by the monomials of degree
(
). One can also define a graded algebra
in a more general manner as an algebra whose additive group can be represented as a direct sum of groups
, where
runs through a certain commutative semi-group
and
for any
. The concept of a filtered algebra is closely connected with that of a graded algebra. It is in fact possible to define, in a natural manner, an ascending filtration
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on each graded algebra . Conversely, if
is a filtered algebra
, then one defines the graded algebra
(where
,
), which is called the graded algebra associated with
. A graded ring is defined in a similar manner.
Comments
For an arbitrary group one may define a gradation of type
on an algebra
, i.e.
where each
is an additive subgroup of
and
for all
. The group algebra
over the field
, as well as crossed products
defined by a group morphism
and a
-cocycle
, are examples of
-graded algebras (see also Cross product). The use of
-gradations that are not necessarily positive gradations allows one to consider the graded rings associated to
-adic filtrations on a ring
; for an ideal
of
the
-adic filtration is given by a descending chain
, and then
, where
is negatively graded.
References
[a1] | N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , Hermann (1961) pp. Chapt. 3. Graduations, filtrations, et topologies |
[a2] | C. Nâstâsescu, F. van Oystaeyen, "Graded ring theory" , North-Holland (1982) |
Graded algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Graded_algebra&oldid=47108