Commutative Ring Theory

19 Graded Rings and Modules

Hideyuki Matsumura Nagoya University, Nagoya, Japan

In this article all rings are assumed to be commutative with a unit element. In section 1 we shall consider graded rings of type G, where G is an abelian group, and discuss some properties which behave well for all G and also some others which depend on G. section 2 will be a brief summary of some of the main results of Goto-Watanabe [1978a]. In section 3 we shall consider the following topics: projective embeddings, graded modules defined by regular differential forms, the canonical ring of a curve.

1. G-graded rings

Let G be an arbitrary abelian group. A graded ring of type G (G-graded ring for short) is a ring R with a direct decomposition (as $\mathbb{Z}\text{−module}$)

$$R=\underset{}{{\displaystyle \oplus }}$$