Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (I)

It is shown that the integral closure R' of a local (Noetherian) domain R is equal to the intersection of the Rees valuation rings of all proper ideals in R of the form (b, I_k)R, where b is an arbitrary nonzero nonunit in R and the I_k are an arbitrary descending sequence of ideals (varying with b and with I_k ⊆ (I_k-1 ∩ I₁^k) for all k > 1, one sequence for each b). Also, this continues to hold when b is restricted to being irreducible and no two distinct b are associates. We prove similar results for a Noetherian domain.

Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)

<p>Let 1 &lt; s₁ &lt; . . . &lt; s_k be integers, and assume that κ ≥ 2 (so s_k ≤ 3). Then there exists a local UFD (Unique Factorization Domain) (R,M) such that:</p><p>(1) Height(M) = s_k.</p><p>(2) R = R' = ∩{VI (V,N) € V_j}, where V_j (j = 1, . . . , κ) is the set of all of the Rees valuation rings (V,N) of the M-primary ideals such that trd((V I N) I (R I M)) = s_j - 1.</p><p>(3) With V₁, . . . , V_κ as in (2), V₁ ∪ . . . V_κis a disjoint union of all of the Rees valuation rings of allof the M-primary ideals, and each M-primary ideal has at least one Rees valuation ring in each V_j .</p>

SOME RESULTS ON ONE-FIBERED IDEALS

The aim of this paper is to prove some results about the one-fibered ideals (i.e. ideals with only one Rees valuation). We will introduce and study the notion of nth roots of this type of ideals and we will present several properties related to its integral closure by using the one-fiberedness criterion of Hübl and Swanson.