S-Noetherian Properties of Composite Ring Extensions

Let R be a commutative ring with identity and S a multiplicative subset of R. We say that R is an S-Noetherian ring if for each ideal I of R, there exist an s ∈ S and a finitely generated ideal J of R such that sI ? J ? I. In this article, we study transfers of S-Noetherian property to the composite semigroup ring and the composite generalized power series ring.     

Universal Lying-Over Rings

A (commutative unital) ring R is said to satisfy universal lying-over (ULO) if each injective ring homomorphism R → T satisfies the lying-over property. If R satisfies ULO, then R = tq(R), the total quotient ring of R. If a reduced ring satisfies ULO, it also satisfies Property A. If a ring R = tq(R) satisfies Property A and each nonminimal prime ideal of R is an intersection of maximal ideals, R satisfies ULO. If 0 ≤ n ≤ ∞, there exists a reduced (resp., nonreduced) n-dimensional ring satisfying ULO. The A + B construction is used to show that if 2 ≤ n < ∞, there exists an n-dimensional reduced ring R such that R = tq(R), R satisfies Property A, but R does not satisfy ULO.     

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 ? ?₀, 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].