Throughout is a group, is a commutative ring with 1 and is a –algebra.

**Definition 1**. is called a –**graded** -algebra if for every there exists a -module such that

1) for all

2) as -modules.

A **homogeneous** element of is any element of If then is called homogeneous of **degree** If then is written uniquely as where and all but finitely many of are zero. Each is called a **homogeneous component** of

**Definition 2**. We say that is **positively graded** if is -graded and for all

The concept of graded algebras is just a very natural generalization of polynomial algebras.

**Example 1**. Let be a -algebra and consider the polynomial algebra Then is positively graded because clearly where Note that This grading for is called the standard grading of

**Example 2**. Let the polynomial algebra in the indeterminates and Let be the set of all polynomials of total degree For example, etc. Then Note that So is positively graded and we call this the standard grading of In genral, the standard grading of the polynomial algebra in the indeterminates is defined by where is the set of polynomials of total degree

**Example 3**. The standard grading of the Laurent polynomial algebra is

**Remark**. Let be a (positively) graded algebra and let Then

1) is a subring of and is an -module for all

2)

3) is a two-sided ideal of

Proof. 1) By the property 1) in Definition 1, we have and That means is both left and right -module. Also, and so is a subring of To prove 2), let where and only a finitely many of are non-zero. We need to show that for all To prove this, let Then But and if then Thus for all and Thus if then Part 3) of the remark is trivial.

**Definition 3**. The ideal in the above remark is called the **augmentation ideal** of

**Example 4**. Let be the polynomial algebra in Example 1. Then the ideal generated by Similarly, the augmentation ideal of the polynomial algebra (see Example 2) is

Excellent.