This tag is for questions relating to "Graded Module", extensively used in homological algebra. 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 Z-algebra.
Graded modules, and in general the concept of grading in algebra, are an essential tool in the study of homological algebraic aspect of rings.
Definition: Let $~R=R0⊕R1⊕⋯~$ be a graded ring. A module $M$ over $R$ is said to be a graded module if $$M=M0⊕M1⊕⋯$$ where $Mi$ are abelian subgroups of $M$, such that $~RiMj⊆Mi+j ~~~~\forall i,j~$.
An element of $M$ is said to be homogeneous of degree $i$ if it is in $Mi$. The set of $Mi$ is called a grading of $M$.
- Whenever we speak of a graded module, the module is always assumed to be over a graded ring.
- As any ring $R$ is trivially a graded ring $($where $Ri=R$ if $i=0$ and $Ri=0$ otherwise$)$, every module $M$ is trivially a graded module with $Mi=M$ if $i=0$ and $Mi=0$ otherwise.
- A graded module (or a graded ring) non-trivially.
- If $R$ is a graded ring, then clearly it is a graded module over itself, by setting $Mi=Ri~~ (M=R$ in this case$)$. Furthermore, if $M$ is graded over $R$, then so is $Mz$ for any indeterminate $z$.
- A graded module that is also a graded ring is called a graded algebra.
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