Completions of UFDs with Semi-Local Formal Fibers

Let (T, M) be a complete local ring such that |T/M| = | T |. Given a finite set of incomparable nonmaximal prime ideals C of T, we provide necessary and sufficient conditions for T to be the completion of a local UFD A, whose generic formal fiber is semilocal with maximal ideals the elements of C. We also show that, given the T above, we can find necessary and sufficient conditions for T to be the completion of a UFD, whose formal fiber over a height one prime ideal is semilocal.

Communicated by I. Swanson.     

Dimensions of Formal Fibers of Height One Prime Ideals

Let T be a complete local (Noetherian) ring with maximal ideal M, P a nonmaximal ideal of T, and C = {Q ₁, Q ₂,…} a (nonempty) finite or countable set of nonmaximal prime ideals of T. Let {p ₁, p ₂,…} be a set of nonzero regular elements of T, whose cardinality is the same as that of C. Suppose that p _i  ∈ Q _j if and only if i = j. We give conditions that ensure there is an excellent local unique factorization domain A such that A is a subring of T, the maximal ideal of A is M ∩ A, the (M ∩ A)-adic completion of A is T, and so that the following three conditions hold: (1) p _i  ∈ A for every i; (2) A ∩ P = (0), and if J is a prime ideal of T with J ∩ A = (0), then J ? P or J ? Q _i for some i; (3) for each i, p _i A is a prime ideal of A, Q _i ∩ A = p _i A, and if J is a prime ideal of T with J ? Q _i , then J ∩ A ≠ p _i A.     

COMMUTATIVE RINGS WITH FINITE QUOTIENT FIELDS

Abstract

We consider the class of all commutative reduced rings for which there exists a finite subset T ? A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for this class of rings,and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals.     

Chains of Rings with Local Formal Fibers

Let (T, M) be a complete local domain containing the integers. Let p ₁ ? p ₂ ? ··· ? p _n be a chain of nonmaximal prime ideals of T such that T _{p n} is a regular local ring. We construct a chain of excellent local domains A _n  ? A _n?1 ? ··· ? A ₁ such that for each 1 ≤ i ≤ n, the completion of A _i is T, the generic formal fiber of A _i is local with maximal ideal p _i , and if I is a nonzero ideal of A _i then A _i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht _T Q, then there is a chain of excellent local domains B ₀ ? B ₁ ? ··· ? B _h  ? T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B _i ) = i, the completion of B _i is isomorphic to T[[X ₁, X ₂,…, X _i ]] where the X _j 's are indeterminants, and the formal fiber of Q ∩ B _i is local.     

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by A_T the localised ring and let M be a primitive ideal of A_T Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QR_Q is generated by a regular g-centralising set of elements, then

(1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (R_Q 0 + ∣X_A (P)∣

(2)d gldim(A_T ) = Kdim(A_T ) = ht(M) = ht(P).     

ABELIAN EXCHANGE MODULES

ABSTRACT

Let M_k be a right k-module with endomorphism ring E=End(M_k). We prove that if E is an Abelian exchange ring, then M_k has the full exchange property. We also give an extension of this result in the case E is regular.     

P. Hall's covering group and embeddings of countable cc-groups

Let A be an elementary abelian group of order at least p ³ acting on a finite p′-group G that is soluble with derived length d. Assume that γ _c (C _G (a)) has exponent dividing m for any a ∈ A ^#. It is proved that there exist {p, d, c, m}-bounded numbers c ₁ and m ₁ such that γ _{c 1} (G) has exponent dividing m ₁.     

On the Witt Vectors of Perfect Rings in Positive Characteristic

The purpose of this article is to prove some results on the Witt vectors of perfect F _p -algebras. Let A be a perfect F _p -algebra for a prime integer p, and assume that A has the property P. Then does the ring of Witt vectors of A also have P? A main theorem gives an affirmative answer for P = ″integrally closed” under a very mild condition.     

Hamilton cycles,avoiding prescribed arcs,in close-to-regular tournaments

The local irregularity of a digraph D is defined as i_l(D) = max {|d⁺ (x) − d⁻ (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V₁, V₂, …, V_c} be a partition of V(T) such that |V₁| ≥ |V₂| ≥ … ≥ |V_c|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2i_l(T) + 2|V₁| + 2|V₂| − 2, i_l(T) + 3|V₁| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., i_l(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999     

On generalized stable rings

In this paper, we introduce the class of generalized stable rings and investigate equivalent characterizations of such rings. We show that End_R R is a generalized stable ring if and only if any right H-module decompositions M = A ₁ + B ₁ = A ₂ + B ₂ with A ₁ ? A ? A ₂ implies that there exist C,D,E ≤ M such that M = C + D B ₁ = c + E + B ₂ with.C ? A. Also we show that every generalized stable ring is a GE-ring and matrices over generalized stable regular ring can be diagonalized by some weakly invertible matrices.     

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S?M?Xreal analytic submonifolds with codium ^R _MS=1,and let ω be a connected component of M\S. Let p∈S X_MT_M ^*X where T^* _Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *.

Under the assumption dim^R(T_pT_M ^*X? ν(p)) = 1, a theorem of vanishing for the cohomology groups H^jμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M.

Under the hypothesis dim^R(T_pT_S ^*X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.     

Saturations of Submodules

Abstract

Let N be a submodule of a module M over a commutative ring R such that M/N is finitely generated. It is shown that a submodule of M is a prime submodule minimal over N if and only if it is the saturation of N + pM for certain prime ideal p of R. The bearing of this result upon the M-radical of N is discussed.     

Multiplicative Lie Isomorphisms Between Prime Rings

Let ? be a prime ring with 1 containing a nontrivial idempotent E, and let ?′ be another prime ring. If Φ:? → ?′ is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z′ _T,S for all T, S ∈ ?, where Z′ _T,S is an element in the center 𝒵′ of ?′ depending on T and S.     

Gaussian Polynomials and Content Ideal in Pullbacks

This article deals mainly with rings (with zerodivisors) in which regular Gaussian polynomials have locally principal contents. Precisely, we show that if (T,M) is a local ring which is not a field, D is a subring of T/M such that qf(D) = T/M, h: T → T/M is the canonical surjection and R = h ^?1(D), then if T satisfies the property “every regular Gaussian polynomial has locally principal content,” then also R verifies the same property. We also show that if D is a Prüfer domain and T satisfies the property “every Gaussian polynomial has locally principal content”, then R satisfies the same property. The article includes a brief discussion of the scopes and limits of our result.     

On Strongly Essential Submodules

The submodules with the property of the title (N ? M is strongly essential in M if ∏ _I N is essential in ∏ _I M for any index set I) are introduced and fully investigated.

It is shown that for each submodule N of M there exists a subset T ? M such that N + T is strongly essential submodule of M and (N:T) = Ann(T), T  ∩  N = 0. Basic properties of these objects and several examples are given and the counterparts of the related concepts to essential submodules are also introduced and studied. It is shown that each maximal left ideal of a left fully bounded ring is either a summand or strongly essential. Rings over which no module has a proper strongly essential submodule are characterized. It is also shown that the left Loewy rings are the only rings over which the essential submodules and strongly essential submodules of any left module coincide. Finally, a new characterization of left FBN rings is observed.     

An application of the cotangent complex to regularity and complete intersection

Let A be a field and C a noetherian A-algebra such that C _q is a quotient ring of an equicharacteristic local regular ring for each prime ideal q of C We prove that if C is geometrically reduced over A and Ω _C/A is flat, then C is regular. We apply this result to prove that a reduced ring homomorphism u :A→C is regular when Ω _C/A is a flat C-module and C is a complete noetherian local ring or a Nagata local ring. We also study the property of C being locally a complete intersection ring when the flat dimension of Ω _C/A finite.     

On linear operators strongly preserving invariants of Boolean matrices

Let U _k be the general Boolean algebra and T a linear operator on M _m,n (U _k ). If for any A in M _m,n (U _k ) (M _n (U _k ), respectively), A is regular (invertible, respectively) if and only if T(A) is regular (invertible, respectively), then T is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over U _k . Meanwhile, noting that a general Boolean algebra U _k is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertible, respectively) matrices over 169-7 _k from another point of view.     

On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs

Let be a strongly regular graph with adjacency matrix A. Let I be the identity matrix, and J the all-1 matrix. Let p be a prime. Our aim is to study the p-rank (that is, the rank over , the finite field with p elements) of the matrices M = aA + bJ + cI for integral a, b, c. This note is based on van Eijl [8].     

Quasicentralizer of LCM-Homomorphisms

Let A _S and A _T be Artin–Tits groups and let ? _p : A _S  →  A _T be an LCM -homomorphism constructed using the folding method. We characterize the quasicentralizer of ? _p (S) in A _T in the case when T is of spherical type.