Pseudo-Dedekind domains and divisorial ideals in R[X]T

On generalized Dedekind prime rings

“…In Section 5 Theorem 5.4, we prove that if R is a PI G-Dedekind prime ring then R[x] is also a PI G-Dedekind prime ring. In the commutative case, Anderson and Kang [3] and Zafrullah [19] proved that if R is a G-Dedekind domain then R[x] is also a G-Dedekind domain by using the lemma of Dedekind-Mertens (see [10]). Since every commutative ring is a PI-ring and we use a different method, the proof of Theorem 5.4 can also be considered as an alternative proof for this result in commutative Noetherian case.…”

“…Anderson and Kang [3], and Zafrullah [19] studied the question: if R is a commutative integral domain when is (AB) −1 = A −1 B −1 for all non-zero fractional ideals A and B of R? ( 1 ) Authors call commutative integral domains satisfying (1), pseudo-Dedekind domains and G-Dedekind domains, respectively.…”

(t, v)-Dedekind Domains and the Ring $${{R[X]_{N_v}}}$$

“…Let X be an indeterminate over R and {X λ } λ∈Λ (or denoted by {X λ }) be an arbitrary set of indeterminates [19] and [20]) Let I be a nonzero fractional ideal of R, then [5,Theorem 4.1] showed that if R is pseudo-Dedekind, then R [X] T is a pseudo-Dedekind domain for any multiplicative closed subset T ⊆ N v . We can extend this property to a (t, v)-Dedekind domain.…”

“….). Anderson and Kang [5,Theorem 4.5] showed that R[X; G] is pseudo-Dedekind domain if and only if R is a pseudo-Dedekind domain and G has type (0, 0, . .…”

Holomorphy rings of function fields

“…An integral domain for which every nonzero divisorial ideal is invertible is termed a "generalized Dedekind domain" in [Zaf86] and a "pseudo-Dedekind domain" in [AK89]. The ring of entire functions is such a Prüfer domain [Zaf86], while the ring of integer-valued polynomials on Z is not [Lop97], though like the holomorphy ring H and the ring of entire functions it is completely integrally closed.…”