**Graduations:**

**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

and

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

**Filtrations:**

**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

**Induced Graduations:**

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

[1] Lang, Serge. *Algebra*. Revised Third Edition. Springer-Verlag. 2000.

[2] Dummit, David and Richard Foote. *Abstract Algebra.* Third Edition. John Wiley and Sons, Inc. 2004.