Editing Graded ring (section)

== Graded module ==
The corresponding idea in [[module theory]] is that of a '''graded module''', namely a left [[module (mathematics)|module]] ''M'' over a graded ring ''R'' such that
: <math>M = \bigoplus_{i\in \mathbb{N}}M_i ,</math>
and
: <math>R_iM_j \subseteq M_{i+j}</math>
for every {{mvar|i}} and {{mvar|j}}.

Examples:
* A [[graded vector space]] is an example of a graded module over a [[field (mathematics)|field]] (with the field having trivial grading).
* A graded ring is a graded module over itself. An ideal in a graded ring is homogeneous [[if and only if]] it is a graded submodule. The [[annihilator (ring theory)|annihilator]] of a graded module is a homogeneous ideal.
* Given an ideal ''I'' in a commutative ring ''R'' and an ''R''-module ''M'', the direct sum <math diaply=inline>\bigoplus_{n=0}^{\infty} I^n M/I^{n+1} M</math> is a graded module over the associated graded ring <math display=inline>\bigoplus_0^{\infty} I^n/I^{n+1}</math>.

A ''morphism'' <math>f: N \to M</math> of graded modules, called a '''graded morphism''' or ''graded homomorphism'' , is a [[Module homomorphism|homomorphism]] of the underlying modules that respects grading; i.e., {{tmath|1= f(N_i) \subseteq M_i }}. A '''graded submodule''' is a submodule that is a graded module in own right and such that the set-theoretic [[inclusion map|inclusion]] is a morphism of graded modules. Explicitly, a graded module ''N'' is a graded submodule of ''M'' if and only if it is a submodule of ''M'' and satisfies {{tmath|1= N_i = N \cap M_i }}. The [[kernel (algebra)|kernel]] and the [[image (mathematics)|image]] of a morphism of graded modules are graded submodules.

Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the [[center (ring theory)|center]] is the same as to give the structure of a graded algebra to the latter ring.

Given a graded module <math>M</math>, the <math>\ell</math>-twist of <math>M</math> is a graded module defined by <math>M(\ell)_n = M_{n+\ell}</math> (cf. [[Serre's twisting sheaf]] in [[algebraic geometry]]).

Let ''M'' and ''N'' be graded modules. If <math>f\colon M \to N</math> is a morphism of modules, then ''f'' is said to have degree ''d'' if <math>f(M_n) \subseteq N_{n+d}</math>. An [[exterior derivative]] of [[differential form]]s in [[differential geometry]] is an example of such a morphism having degree 1.