Commutative graded-n-coherent and graded valuation rings

Let R = circle plus R-alpha is an element of G(alpha) be a commutative ring with unity graded by an arbitrary grading commutative monoid G. For each positive integer, the notions of a graded-n-coherent module and a graded-n-coherent ring are introduced. In this paper many results are generalized from n-coherent rings to graded-n-coherent rings. In the last section, we provide necessary and sufficient conditions for the graded trivial extension ring to be a graded-valuation ring.