Graded ring

Definition
A graded ring is a commutative unital ring $$A$$ equipped with a direct sum decomposition as a sum of Abelian subgroups:

$$A = \oplus_{i=-\infty}^\infty A_i = \cdots A_{-2} \oplus A_{-1} \oplus A_0 \oplus A_1 \oplus A_2 \oplus \cdots$$

such that the following hold:


 * Each $$A_i$$ is a subgroup under addition
 * $$1 \in A_0$$
 * $$A_mA_n \subset A_{m+n}$$. In other words, if $$a \in A_m$$ and $$b \in A_n$$ then $$ab \in A_{m+n}$$

A structure of the above sort on a ring is termed a gradation, also a $$\mathbb{Z}$$-gradation. The ring $$A$$ is positively graded if $$A_i = 0$$ for all $$i<0$$.

There are related notions for noncommutative rings.

Weaker notions

 * Filtered ring