Valuation ideals and primary w-ideals

Abstract

Let D be an integral domain, V(D) (resp., t-V(D)) be the set of all valuation (resp., t-valuation) ideals of D, and w-P(D) be the set of primary w-ideals of D. Let D[X] be the polynomial ring over D, c(f) be the ideal of D generated by the coefficients of f ∈ D[X], and N_v= {f ∈ D[X] | c(f)_v=D}. In this paper, we study integral domains D in which w-P(D) ⊆ t-V(D), t-V(D) ⊆ w-P(D), or t-V(D) = w-P(D). We also study the relationship between t-V(D) and V({D[X]}_{N_v}), and characterize when t-V(A + XB[X]) ⊆w-P(A + XB[X]) holds for a proper extension A ⊂ B of integral domains.

Keywords

t-Valuation ideal / primary w-ideal / PvMD / UMT-domain / D[X]_Nv

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2016 Higher Education Press and Springer-Verlag Berlin Heidelberg