On Graded Dedekind Domains

In this paper, we study the structure theory of graded Dedekind domains andshow that:（1） Let M be a graded Noether module. Then every proper graded submodule of M hasgraded primary decomposition.（2）（The Krull intersection theorem on graded Noether rings） Let I be a homogeneous∞ideal of a graded Noether ring R and let M be a graded R-module. If B=∩|∞IⁿM, thenB=IB.（3） R is a graded Dedekind domain if and only if R is a graded Noether ring and Rpisgraded discrete valuation rings for every homogeneous prime ideal p of R and if and only ifevery homogeneous prime ideal of R is invertible.（4） Not only the graded local Dedekind domains are graded discrete valuation rings, butalso are graded principal ideal rings.（5） If R is a graded Dedekind domain, then （a） the ABLK theory is established;（b）every nonzero homogeneous fractional ideal of R can be generated by two homogeneouselements;（c） every nonzero homogeneous fractional ideal of R can be factored uniquely asP₁ⁿ¹P₂ⁿ²Â·Â·Â·P_r^nr;（d） every graded primary ideal of R is a power of graded prime ideal.