Definition:Z-Graded Ring

Definition
Let $\Z$ be the set of integers.

A $\Z$-graded ring is a graded ring of type the additive group of integers.

That is, it is a pair $(R, f)$ where $R$ is a ring and $f$ is a family $(R_n)_{n \in \Z}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that:
 * $\forall x \in R_n, y \in R_m: x y \in R_{m+n}$

Also see

 * Definition:N-Graded Ring