## 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.