Francesca Tartarone

When the Semistar Operation is the Identity

We study properties of integral domains in which it is given a semistar operation ★ such that is the identity. In particular, we put attention to the case ★ = v, where v is the divisorial closure.

STAR-INVERTIBILITY AND t-FINITE CHARACTER IN INTEGRAL DOMAINS

Let A be an integral domain. We study new conditions on families of integral ideals of A in order to get that A is of t-finite character (i.e. each nonzero element of A is contained in finitely many t-maximal ideals). We also investigate problems connected with the local invertibility of ideals.

Strong Mori and Noetherian properties of integer-valued polynomial rings

Abstract Let D be a domain with quotient field K and let Int( D ) be the ring of integer-valued polynomials {f∈K[X] | f(D)⊆D} . We give conditions on D so that the ring Int( D ) is a Strong Mori domain. In particular, we give a complete characterization in the case that the conductor (D : D′) is nonzero, where D ′ is the integral closure of D . We also show that when D is quasilocal with Int (D) ≠ D[X] or D is Noetherian, Int( D ) is a Strong Mori domain if and only if Int( D ) is Noetherian.

Polynomial Closure in Noetherian Pseudo-valuation Domains

Abstract Let D be a domain with quotient field K and E ⊆ K be a subset. We consider the ring Int(E, D) ≔ {f ∈ K[X]; f(E) ⊆ D} of integer-valued polynomials over E . The polynomial closure of E is cl D (E) ≔ {x ∈ K; f(x) ∈ D, ∀ f ∈ Int(E, D)}. We study cl D (I), when I is a fractional ideal of a Noetherian pseudo-valuation domain.

On the class group of integer-valued polynomial rings over Krull domains

Let D be a Krull domain with quotient field K. We study the class group of the integer-valued polynomial ring over D, Int(D)≔{f∈K[X];f(D)⊆D}. In particular, we give necessary and sufficient conditions on D for the class group of Int(D) to be generated by the classes of the t-invertible t-prime ideals and, in this case, we describe its generators. A case of particular interest is when D is a UFD. We also characterize Krull domains D for which Int(D) is a GCD-domain.

Divisorial prime ideals of int(D) when D is a krull-type domain

Let Dbe a domain with quotient field K. The ring of integer-valued polynomials over Dis Int (D):={fϵ K[X];f(4D) ⊆ D}.We describe the divisorial prime ideals of Int(D) when Dis a domain of Krull-type and, in particular, when Dis also a d-ring.

Integrally closed rings in birational extensions of two-dimensional regular local rings

Let

Invertibility of ideals in Prüfer extensions

D

Integer-Valued Polynomials and Prüfer v-Multiplication Domains☆

be an integrally closed local Noetherian domain of Krull dimension 2, and let

Flat ideals and stability in integral domains

f