Homogeneous star-operations on graded integral domains

Let Γ be a torsion-free cancellative monoid and let R=⊕α∈ΓRα be a graded integral domain. In this paper, we define the concept of homogeneous star-operations, which is a generalization of that of star-operations, and apply some properties of star-operations. Also, we define the concept of homogeneous divisor classes, which is the graded integral domain version of that of divisor classes, and study some equivalent conditions for the set of homogeneous divisor classes HD(R) of R to be a group.