Definition 1. Let be a monoid. An –graded ring is a ring where
and where are abelian groups. A graded ring will mean an -graded ring. is an –graded –algebra if it is -graded as a ring. If is a graded -algebra and is graded, then we say is graded with respect to if and
Example 2. The tensor algebra of a module is trivially graded with its concatenation product. A group ring is a -graded ring, which can be seen with its decomposition A -graded ring (algebra) is called a super ring (super algebra). Note that -graded rings and graded rings induce a grading as follows
Note we could slightly generalize the above definition of being graded with respect to as follows: suppose is -graded and is -graded for monoids and that there exists a monoid homomorphism Then we can say that is an –graded –algebra if
A similar definition exists for an -graded -module. Note an -graded -algebra has an -grading where is the trvial monoid grading and is of course the trivial morphism. An element of an -graded -algebra is called homogeneous (of order ) if it has the form for some Let be -graded -algebras and be an -algebra homomorphism. Then is called a graded homomorphism if In more generality, we could have be an -graded -algebra together with a module homomorphism and require
Definition 3. A subset is homogeneous if for every element of the component elements (which are homogeneous) are in If is an ideal, then we call it a homogeneous (or graded) ideal. Note it has a grading as it can be written as a direct sum of the ideals generated by the homogeneous elements.
Proposition 4. Let be an -graded -algebra and be a homogeneous ideal in Then is an -graded -algebra.
Proof. Since is homogeneous, it is graded (since elements give component elements). So we have
Cosets of in the quotient thus have the form
which gives us a decomposition. Hence
Definition 5. Let be a poset. A subset is called a filter if the following hold
- with and
- If and then
Definition 6. Let be a structure and be a filter. An descending (ascending) filtration on with respect to is a collection of substructures of such that (). A filtration on will mean a filtration with respect to the filter (either ascending/descending). We can similarly define filtrations of modules and filtrations of modules that respect the filtration of their ring.
Definition 7. If is an ideal in and is an -module with a descending filtration, then the filtration is called an –filtration if for all It is called –stable if for all for some
Hence we can view multiplication by as increasing/decreasing the degree (depending upon preferred terminology) of elements in
Let be an ideal of Then has an -filtration:
We can define the Rees algebra (which Lang calls the “first associated graded ring”) as
This is clearly a graded -algebra. We could also consider
This is easily verified as a graded -algebra with a product defined componentwise:
Definition 8. Let be a graded -module with grading We define the Hilbert polynomial by We define the of as
 Lang, Serge. Algebra. Revised Third Edition. Springer-Verlag. 2000.
 Dummit, David and Richard Foote. Abstract Algebra. Third Edition. John Wiley and Sons, Inc. 2004.