Irredundant intersections of valuation overrings of two-dimensional Noetherian domains

Let D be a two-dimensional Noetherian domain, let R be an overring of D, and let Σ and Γ be collections of valuation overrings of D. We consider circumstances under which (_VΣV)∩R=(_WΓW)∩R implies that Σ=Γ. We show that if R is integrally closed, these representations are “strongly” irredundant, and every member of ΣΓ has Krull dimension 2, then Σ=Γ. If in addition Σ and Γ are Noetherian subspaces of the Zariski–Riemann space of the quotient field of D (e.g. if Σ and Γ have finite character), then the restriction that the members of ΣΓ have Krull dimension 2 can be omitted. An example shows that these results do not extend to overrings of three-dimensional Noetherian domains.     

Strong Mori and Noetherian properties of integer-valued polynomial rings

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 is nonzero, where D′ is the integral closure of D. We also show that when D is quasilocal with or D is Noetherian, Int(D) is a Strong Mori domain if and only if Int(D) is Noetherian.     

Indecomposable modules over one-dimensional Noetherian rings

In this article we consider finitely generated torsion-free modules over certain one-dimensional commutative Noetherian rings R. We assume there exists a positive integer N_R such that, for every indecomposable R-module M and for every minimal prime ideal P of R, the dimension of M_P, as a vector space over the field R_P, is less than or equal to N_R. If a nonzero indecomposable R-module M is such that all the localizations M_P as vector spaces over the fields R_P have the same dimension r, for every minimal prime P of R, then r=1,2,3,4 or 6. Let n be an integer ≥8. We show that if M is an R-module such that the vector space dimensions of the M_P are between n and 2n−8, then M decomposes non-trivially. For each n≥8, we exhibit a semilocal ring and an indecomposable module for which the relevant dimensions range from n to 2n−7. These results require a mild equicharacteristic assumption; we also discuss bounds in the non-equicharacteristic case.     

Ratliff-Rush closures of ideals with respect to a Noetherian module

Let R be a commutative Noetherian ring, E a non-zero finitely generated R-module and I a E-proper ideal of R. The purpose of this paper is to provide some new characterizations of when all powers of I are Ratliff-Rush closed with respect to E and to answer a question raised by W. Heinzer et al. in (The Ratliff-Rush Ideals in a Noetherian Ring: A Survey, in Methods in Module Theory, Dekker, New York, 1992, pp. 149-159).     

Ideals contracted from a Noetherian extension ring

Let R be a commutative ring with identity such that for each ideal A of R, there exists a Noetherian unitary extension ring T(A) of R such that A is contracted from T(A). We investigate the structure of R. The context in which this topic has usually been considered is where R is an integal domain and T(A) is an overring of R. Under these hypotheses we show that R is Neotherian if R is one-dimensional. In the general case, R is strongly Laskerian, has Noetherian spectrum, and satisfies certain chain conditions for quotient ideals, but R need not be Noetherian.     

On one class of modules that are close to Noetherian

We consider an R G-module A over a commutative Noetherian ring R. Let G be a group having infinite section p-rank (or infinite 0-rank) such that C _G (A) = 1, A/C _A (G) is not a Noetherian R-module, but the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank, respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.     

On the set of <Emphasis Type="Italic">t</Emphasis>-linked overrings of an integral domain

Let R be an integral domain with quotient field L. An overring T of R is t-linked over R if I ⁻¹ = R implies that (T : IT)  =  T for each finitely generated ideal I of R. Let O _t (R) denotes the set of all t-linked overrings of R and O(R) the set of all overrings of R. The purpose of this paper is to study some finiteness conditions on the set O _t (R). Particularly, we prove that if O _t (R) is finite, then so is O(R) and O _t (R) = O(R), and if each chain of t-linked overrings of R is finite, then each chain of overrings of R is finite. This yields that the t-linked approach is more efficient than the Gilmer’s treatment (Proc Am Math Soc 131:2337–2346, 2002). We also examine the finiteness conditions in some Noetherian-like settings such as Mori domain, quasicoherent Mori domain, Krull domain, etc. We establish a connection between O _t (R) and the set of all strongly divisorial ideals of R and we conclude by a characterization of domains R that are t-linked under all their overrings. This work was funded by KFUPM under Project # FT/18-2005.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Unique representation domains, II

Given a star operation ∗ of finite type, we call a domain R a ∗-unique representation domain (∗-URD) if each ∗-invertible ∗-ideal of R can be uniquely expressed as a ∗-product of pairwise ∗-comaximal ideals with prime radical. When ∗ is the t-operation we call the ∗-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah [M. Zafrullah, On unique representation domains, J. Nat. Sci. Math. 18 (1978) 19-29] and Brewer-Heinzer [J.W. Brewer, W.J. Heinzer, On decomposing ideals into products of comaximal ideals, Comm. Algebra 30 (2002) 5999-6010], we give conditions for a ∗-ideal to be a unique ∗-product of pairwise ∗-comaximal ideals with prime radical and characterize ∗-URD’s. We show that the class of URD’s includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction.     

The Ideal Completion of a Noetherian Local Domain

The ideal topology on a integral domain R is the linear topology which has as a fundamental system of neighborhoods of 0 the nonzero ideals of R. We investigate the properties of the ideal topology on a Noetherian local domain (R, 𝔪), and we establish connections between the 𝔪-adic completion and the ideal completion. We give conditions under which the completion in the ideal topology is Noetherian, and we show that, unlike the 𝔪-adic completion, the completion in the ideal topology is not always Noetherian.     

Global Theory of Lattice-Finite Noetherian Rings

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal. Presented by H. Tachikawa     

Lazy bases: a minimalist constructive theory of Noetherian rings

We give a constructive treatment of the theory of Noetherian rings. We avoid the usual restriction to coherent rings; we can even deal with non‐discrete rings. We introduce the concept of rings with certifiable equality which covers discrete rings and much more. A ring R with certifiable equality can be fitted with a partial ideal membership test for ideals of R. Lazy bases of ideals of R [X ] are introduced in order to derive a partial ideal membership test for ideals of R [X ]. It is then proved that if R is Noetherian, then so is R [X ]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Prescribed subintegral extensions of local Noetherian domains

We show how subintegral extensions of certain local Noetherian domains S$S$ can be constructed with specified invariants including reduction number, Hilbert function, multiplicity and local cohomology. The construction behaves analytically like Nagata idealization but rather than a ring extension of S$S$, it produces a subring R$R$ of S$S$ such that R⊆S$R\subseteq S$ is subintegral.     

Gorenstein injective, Gorenstein flat modules and the section functor

Let R be a commutative Noetherian ring of Krull dimension d, and let a be an ideal of R. In this paper, we will study the strong cotorsioness and the Gorenstein injectivity of the section functor Γ_a(−) in local cohomology. As applications, we will find new characterizations for Gorenstein and regular local rings. We also study the effect of the section functors Γ_a(−) and the functors on the Auslander and Bass classes.     

Homological properties of the perfect and absolute integral closures of Noetherian domains

For a Noetherian local domain R let R ⁺ be the absolute integral closure of R and let R _∞ be the perfect closure of R, when R has prime characteristic. In this paper we investigate the projective dimension of residue rings of certain ideals of R ⁺ and R _∞. In particular, we show that any prime ideal of R _∞ has a bounded free resolution of countably generated free R _∞-modules. Also, we show that the analogue of this result is true for the maximal ideals of R ⁺, when R has residue prime characteristic. We compute global dimensions of R ⁺ and R _∞ in some cases. Some applications of these results are given.     

Primes of Gauss Extensions Over Crossed Product Algebras

Let K be a skew field with total subring V and G be a right ordered group with cone P, so that the crossed product algebra K*G has a skew field D of fractions. We consider total subrings R of D with R ∩ K = V, describe the overrings in D, as well as subrings of R. For particular extensions R of V we determine the prime ideals of R in terms of prime ideals of V and prime ideals of overcones of P in G.