Graded finite conductor rings

Let [Formula: see text] be a commutative ring graded by an arbitrary commutative cancellation monoid [Formula: see text]. We say that [Formula: see text] is a graded finite conductor ring if [Formula: see text] and [Formula: see text] are finitely generated for every homogeneous elements [Formula: see text] of [Formula: see text]. In this paper, we generalize several results on finite conductor rings to graded finite conductor rings. We then study the possible transfer of this property to the graded trivial ring extension and the graded amalgamation. Our goal is to provide examples of new classes of [Formula: see text]-graded rings that satisfy the above property.