Valuation ideals and primary w-ideals

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 (Vleft( {D{{left[ X right]}_{{N_v}}}} right)), and characterize when t-V (A + XB[X]) ? w-P(A + XB[X]) holds for a proper extension A ? B of integral domains.