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
 |
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