Strongly etale extensions and weak henseliziations

Abstract

Since the circulation, in 1974, of the first draft of “The construction D + XD _S [X], J. Algebra 53 (1978), 423–439” a number of variations of this construction have appeared. Some of these are: The generalized D + M construction, the A + (X)B[X] construction, with X a single variable or a set of variables, and the D + I construction (with I not necessarily prime). These constructions have proved their worth not only in providing numerous examples and counter examples in commutative ring theory, but also in providing statements that often turn out to be forerunners of results on general pullbacks. The aim of this paper will be to discuss these constructions and the remarkable uses they have been put to. I will concentrate more on the A + XB[X] construction, its basic properties and examples arising from it.     

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.     

The Relative Brauer Group of an Affine Double Plane

We study algebra classes and divisor classes on a normal affine surface of the form z ² = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z ² ? f), and if R = k[x, y][f ^?1] and S = R[z]/(z ² ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H¹(G, Cl(T)) → H¹(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H¹(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples.     

On Generalizations of Prime Ideals

Let R be a commutative ring with identity. Let φ: S(R) → S(R) ∪ {?} be a function, where S(R) is the set of ideals of R. Suppose n ≥ 2 is a positive integer. A nonzero proper ideal I of R is called (n ? 1, n) ? φ-prime if, whenever a ₁, a ₂, ?, a _n  ∈ R and a ₁ a ₂?a _n  ∈ I?φ(I), the product of (n ? 1) of the a _i 's is in I. In this article, we study (n ? 1, n) ? φ-prime ideals (n ≥ 2). A number of results concerning (n ? 1, n) ? φ-prime ideals and examples of (n ? 1, n) ? φ-prime ideals are also given. Finally, rings with the property that for some φ, every proper ideal is (n ? 1, n) ? φ-prime, are characterized.     

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].     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

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.     

On Relative K 2-Group of Split Radical Ideals

Let I be a split radical ideal of a ring R. In this article, the exact sequence 1 → K ₂(R, I) → U _R (I) → V(R, I) → 1 is given by using the method of extension of groups, where U _R (I) is determined by generators and relations. The results of Maazen and Stienstra on the presentation for relative K ₂ group of split radical pairs are extended and amplified.     

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 ? ?₀, 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.     

Monomorphism Operator and Perpendicular Operator

For a quiver Q, a k-algebra A, and an additive full subcategory 𝒳 of A-mod, the monomorphism category Mon(Q, 𝒳) is introduced. The main result says that if T is an A-module such that there is an exact sequence 0 → T _m  → … → T ₀ → D(A _A ) → 0 with each T _i  ∈ add(T), then Mon(Q, ^⊥ T) =^⊥(kQ ? _k T); and if T is cotilting, then kQ ? _k T is a unique cotilting Λ-module, up to multiplicities of indecomposable direct summands, such that Mon(Q, ^⊥ T) =^⊥(kQ ? _k T).

As applications, the category of the Gorenstein-projective (kQ ? _k A)-modules is characterized as Mon(Q, 𝒢𝒫(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q, 𝒳) can be described; and a sufficient and necessary condition for Mon(Q, A) being of finite type is given.