Editing Graded ring (section)

== 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 (mathematics)|ring]] that is decomposed into a [[direct sum]]
: <math>R = \bigoplus_{n=0}^\infty R_n = R_0 \oplus R_1 \oplus R_2 \oplus \cdots</math> 
of 
[[additive group]]s, such that 
: <math>R_mR_n \subseteq R_{m+n}</math>
for all nonnegative integers <math>m</math> and {{tmath|1= n }}.

A nonzero element of <math>R_n</math> is said to be ''homogeneous'' of ''degree'' {{tmath|1= n }}. By definition of a direct sum, every nonzero element <math>a</math> of <math>R</math> can be uniquely written as a sum <math>a=a_0+a_1+\cdots +a_n</math> where each <math>a_i</math> is either 0 or homogeneous of degree {{tmath|1= i }}. The nonzero <math>a_i</math> are the ''homogeneous components'' of&nbsp;{{tmath|1= a }}.

Some basic properties are:
* <math>R_0</math> is a [[subring]] of {{tmath|1= R }}; in particular, the multiplicative identity <math>1</math> is a homogeneous element of degree zero.
* For any <math>n</math>, <math>R_n</math> is a two-sided {{tmath|1= R_0 }}-[[module (mathematics)|module]], and the direct sum decomposition is a direct sum of {{tmath|1= R_0 }}-modules.
* <math>R</math> is an [[associative algebra|associative {{tmath|1= R_0 }}-algebra]].

An [[ideal (ring theory)|ideal]] <math>I\subseteq R</math> is ''homogeneous'', if for every {{tmath|1= a \in I }}, the homogeneous components of <math>a</math> also belong to {{tmath|1= I }}. (Equivalently, if it is a graded submodule of {{tmath|1= R }}; see {{section link||Graded module}}.) The [[intersection (set theory)|intersection]] of a homogeneous ideal <math>I</math> with <math>R_n</math> is an {{tmath|1= R_0 }}-[[submodule]] of <math>R_n</math> called the ''homogeneous part'' of degree <math>n</math> of {{tmath|1= I }}. A homogeneous ideal is the direct sum of its homogeneous parts.

If <math>I</math> is a two-sided homogeneous ideal in {{tmath|1= R }}, then <math>R/I</math> is also a graded ring, decomposed as
: <math>R/I = \bigoplus_{n=0}^\infty R_n/I_n,</math>
where <math>I_n</math> is the homogeneous part of degree <math>n</math> of {{tmath|1= I }}.