Graded ring

In commutative algebra and algebraic geometry, a graduated ring is a generalization of the polynomial ring in several variables. He is in algebraic geometry a means to describe projective varieties.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

Definition

A graduated ring A is a ring that has a representation as a direct sum of abelian groups:

So that

Elements of are called homogeneous elements of degree. Each element of a graded ring can be uniquely written as a sum of homogeneous elements.

An ideal is called homogeneous if:

Is an ideal of the ring, the ring associated to the ideal can be formed:

Properties

• An ideal is then exactly homogeneous, if it can be produced by homogeneous elements.
• The sum, the product, the interface and the radical homogeneous ideals is again homogeneous.
• A homogeneous ideal is precisely then prime, if for all homogeneous:

• Is noetherian and an ideal, then it is also noetherian.

Characterization of regular rings

Is a local noetherian ring, its maximal ideal, and a basis of the vector space, then the following statements are equivalent:

Examples

• When a body is, then is naturally a graded ring.
• This ring may also be provided with another graduation: