Gyu Whan Chang

Uppers to Zero in Polynomial Rings and Prüfer-Like Domains

Let D be an integral domain and X an indeterminate over D. It is well known that (a) D is quasi-Prüfer (i.e., its integral closure is a Prüfer domain) if and only if each upper to zero Q in D[X] contains a polynomial g ∈ D[X] with content c D (g) = D; (b) an upper to zero Q in D[X] is a maximal t-ideal if and only if Q contains a nonzero polynomial g ∈ D[X] with c D (g) v = D. Using these facts, the notions of UMt-domain (i.e., an integral domain such that each upper to zero is a maximal t-ideal) and quasi-Prüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this article, given a semistar operation ☆ in the sense of Okabe–Matsuda, we introduce the ☆-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UMt-domains and the Prüfer v-multiplication domains.

Integral Domains in which Every Nonzero t-Locally Principal Ideal is t-Invertible

Let D be an integral domain, D[X] be the polynomial ring over D, and w be the so-called w-operation on D. We define D to be a w-LPI domain if every nonzero w-locally principal ideal of D is w-invertible. This article presents some properties of w-LPI domains. It is shown that D is a w-LPI domain if and only if D[X] is a w-LPI domain, if and only if D[X] N v is an LPI domain, where N v = {f ∈ D[X] | c(f)−1 = D}. As a corollary, it is also shown that D is a w-LPI domain if and only if each w-locally finitely generated and w-locally free submodule of D n of rank n over D is of w-finite type. We show that if is a finite character intersection of t-linked overrings D α and if each D α is a w-LPI domain, then D is a w-LPI domain.

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ⊊ ℕ0, D[Γ] be the semigroup ring of Γ over D (and hence D ⊊ D[Γ] ⊊ D[X]), and D + X n K[X] = {a + X n g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X n K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X n K[X].

The class group of integral domains

Abstract Let D be an integral domain, S a saturated multiplicative subset of D, and N={0≠x∈D∣xD∩sD=xsD for all s∈S}. We study Nagatas theorem for the class group and multiplicative sets S such that SN is the complement of a prime ideal. As applications, we calculate the class group of pullback domains and D+XDS[X] domains.

Almost Prüfer v-Multiplication Domains and Related Domains of the Form D + D S [Γ*]

Let D be an integral domain, S be a (saturated) multiplicative subset of D such that D ⊊ D S , Γ be a numerical semigroup with Γ ⊊ ℕ0, Γ* = Γ∖{0}, X be an indeterminate over D, D + XD S [X] = {a + Xg ∈ D S [X]∣a ∈ D and g ∈ D S [X]}, and D + D S [Γ*] = {a + f ∈ D S [Γ]∣a ∈ D and f ∈ D S [Γ*]}; so D + D S [Γ*] ⊊ D + XD S [X]. In this article, we study when D + D S [Γ*] is an APvMD, an AGCD-domain, an AS-domain, an AP-domain, or an AB-domain.

The Class Group of D[Γ] for D an Integral Domain and Γ a Numerical Semigroup

Abstract Let D be an integral domain with quotient field K and Γ a numerical semigroup. We show that Cl(D[Γ]) = Cl(D[X]) ⊕ Pic(K[Γ]), and if D is integrally closed, then Cl(D[Γ]) = Cl(D) ⊕ Pic(K[Γ]).

Semigroup Rings as Weakly Factorial Domains

Let D be an integral domain, Γ be a torsion-free grading monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. We show that if G is of type (0, 0, 0,…), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup. Let ℝ be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers. We also show that Γ is a weakly factorial GCD-semigroup, but ℝ[Γ] is not a weakly factorial domain.

A General Theory of Splitting Sets

ABSTRACT Let * be a finite type star operation on an integral domain D, and let S be a saturated multiplicative subset of D. We say that S is a g*-splitting set of D if for each 0 ≠ d ∈ D, we have d = st for some s ∈ S and t ∈ D with (s′,t)* = D for all s′ ∈ S. In this article, we generalize several well-known results about splitting sets to g*-splitting sets.

THE RINGS

Let D be an integral domain,

D((\mathcal{X}))_i

{\mathcal{X}}