Ideal factorization in strongly discrete independent rings of Krull type

초록

Let D be an integral domain and w be the so-called w-operation on D. In this paper, we define the notion of w-ZPUI domains which is a natural generalization of ZPUI domains introduced by Olberding in 2000. We say that D is a w-ZPUI domain if every nonzero proper w-ideal I of D can be written as I = (JP(1) middot middot middot P-n)w for some w-invertible ideal J of D and {P-1, ... , P-n} is a nonempty collection of pairwise w-comaximal prime w-ideals of D. Then, among other things, we show that D is a w-ZPUI domain if and only if the polynomial ring D[X] is a w-ZPUI domain, if and only if D is a strongly discrete independent ring of Krull type. We construct three types of new w-ZPUI domains from an old one by A + X B[X]-construction, pullback, and D[{X-i, Y-i, U-i, V-i}]/({XiVi-Y(i)Ui(}))-domains. We also show that given an abelian group G, there is a ZPUI domain with ideal class group G but not a Dedekind domain. Finally, we introduce and study the notion of w-ISP domains as a generalization of w-ZPUI domains.

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