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Let n⩾1 and k⩾0 be fixed integers and R be a ring. We show that R is (n, k)-SG with G‐gldim(R)=k if and only if R[x1,x2,…,xm] is (n,k+m)-SG with G‐gldim(R[x1,x2,…,xm])=k+m. We also give an example ...

Let R = (Rn)n≥0 be an ascending chain of commutative rings with identity and H(R) (resp., h(R)) the generalized composite Hurwitz series ring (resp., generalized composite Hurwitz polynomial ring)....

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.

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.

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.

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].