On some graded objects in graded module category

Let [Formula: see text] be a [Formula: see text]-graded ring and let [Formula: see text] be the category of [Formula: see text]-graded [Formula: see text]-modules and homogeneous homomorphisms. In this paper, we define and study some objects in this category. More precisely, we introduce the concepts of graded duo (weak and strong graded duo) modules and give some sources and an example for these types of modules. It is seen that, under some condition, graded duo property is a local property in this category. When the ring [Formula: see text] is a discrete graded valuation ring with unique [Formula: see text]maximal ideal [Formula: see text], we see that these three types of graded (duo) modules are identical and give an explicit characterization of them, so that any graded duo modules over such a ring is of the form [Formula: see text] or [Formula: see text] for some positive integer [Formula: see text] and some integers [Formula: see text]. The same task is done whenever [Formula: see text] is a graded Dedekind domain. Finally, by an example, that provides a wide variety of strong graded duo modules, it was shown that the given characterizations do not hold valid if the ground ring is not Dedekind.