Simple rings are very hard to study because most techniques in ring theory depend on the existence of two-sided ideals. In the case of the Weyl algebra, however, we have a way out. As we saw in Ch. 2, one may define a degree for the elements of the Weyl algebra. Using this degree, we construct a commutative ring, k[x] works as a shadow of An. We may then draw an outline of what An really looks like. This is the best method we have for understanding the structure of An and of its modules.

GRADED RINGS

An important feature of a polynomial ring is that it admits a degree function. We want to generalize and formalize what it means for an algebra to have a degree. This leads to the definition of graded rings. These rings find their justification in algebraic geometry, more precisely in projective algebraic geometry; for details see [Hartshorne, Ch. 1, §2], For the sake of completeness, we define graded rings without assuming commutativity.

Let R be a K-algebra. We say that R is graded if there are K-vector subspaces Ri, i ∈ ℕ, such that

1. (1) R = ⊕i∈ℕRi,

2. (2) Ri · Ri ⊆ Ri+.

The Ri are called the homogeneous components of R. The elements of Ri are the homogeneous elements of degree i. If Ri = 0 when i < 0 then we say that the grading is positive. From now on all graded algebras will have a positive grading unless explicitly stated otherwise.