In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a graded ring is a
ring such that the underlying
additive group is a
direct sum of abelian groups
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
such that
. The index set is usually the set of nonnegative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s or the set of integers, but can be any
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
. The direct sum decomposition is usually referred to as gradation or grading.
A graded module is defined similarly (see below for the precise definition). It generalizes
graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded
-algebra.
The
associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to
non-associative algebras as well; e.g., one can consider a
graded Lie algebra.
First properties
Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article.
A graded ring is a
ring that is decomposed into a
direct sum
:
of
additive groups, such that
:
for all nonnegative integers
and
.
A nonzero element of
is said to be ''homogeneous'' of ''degree''
. By definition of a direct sum, every nonzero element
of
can be uniquely written as a sum
where each
is either 0 or homogeneous of degree
. The nonzero
are the ''homogeneous components'' of
.
Some basic properties are:
*
is a
subring of
; in particular, the multiplicative identity
is an homogeneous element of degree zero.
*For any
,
is a two-sided
-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
, and the direct sum decomposition is a direct sum of
-modules.
*
is an
associative -algebra.
An
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
is ''homogeneous'', if for every
, the homogeneous components of
also belong to
(Equivalently, if it is a graded submodule of
; see .) The
intersection of a homogeneous ideal
with
is an
-
submodule of
called the ''homogeneous part'' of degree
of
. A homogeneous ideal is the direct sum of its homogeneous parts.
If
is a two-sided homogeneous ideal in
, then
is also a graded ring, decomposed as
:
where
is the homogeneous part of degree
of
.
Basic examples
*Any (non-graded) ring ''R'' can be given a gradation by letting
, and
for ''i'' ≠ 0. This is called the trivial gradation on ''R''.
*The
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...