Remark

(1) The graduations most often used are of type $\mathbf{Z}$ or of type $\mathbf{Z}^n$; when we speak of graded (resp. bigraded, trigraded, etc.) modules and rings without mentioning the type, it is understood that we mean graduations of type $\mathbf{Z}$ (resp. $\mathbf{Z}^2, \mathbf{Z}^3$, etc.); a graded ring (resp. module) of type $\mathbf{N}$ is also called a graded ring (resp. module) with positive degrees.

(2) The graded $\mathbf{Z}$-modules of type $\Delta$, when $\mathbf{Z}$ has the trivial graduation, are just the graded commutative groups (whose set of degrees is a commutative monoid) of Definition 1 (no. 1).