An algorithm for the divisors of monic polynomials over a commutative ring

Gilmer and Heinzer proved that given a reduced ring R, a polynomial f divides a monic polynomial in R[X] if and only if there exists a direct sum decomposition of R = R₀ ⊕ … ⊕ R_m (m ≤ deg f), associated to a fundamental system of idempotents e₀, … , e_m, such that the component of f in each R_i[X] has degree coefficient which is a unit of R_i. We propose to give an algorithm to explicitly find such a decomposition. Moreover, we extend this result to divisors of doubly monic Laurent polynomials.     

The endomorphism ring of a Δ-module over a right noetherian ring

LetR be a right noetherian ring. A moduleM _R is called a Δ-module providedR satisfies the descending chain condition for annihilators of subsets ofM. For a Δ-module, a series 0?M ₁?M ₂?...?M _n =M can be constructed in which the factorsM _i /M _i?1 are sums of, α _i -semicritical modules where α₁≦α₂≦...≦α _n . In this paper we utilize this series in studying Λ=End(M _R ). It is shown that ifN={f∈Λ|Kerf is essential inM}, thenN is nilpotent. Specific bounds on the index of nilpotency are given in terms of this series. Further ifM is injective and α-smooth, the annihilators of the factors of this series are used to provide necessary and sufficient conditions for EndM _R to be semisimple.     

Near-rings with no non-zero nilpotent two-sidedR-subsets

It has been proved that, ifR is a near-ring with no non-zero nilpotent two-sidedR-subsets and if the annihilator of any non-zero ideal is contained in some maximal annihilator, thenR is a subdirect sum of strictly prime near-rings. Moreover, ifR is a near-ring with no non-zero nilpotent two-sidedR-subsets and satisfying a.c.c. or d.c.c. on annihilating ideals of the form Ann (Q), whereQ is an ideal ofR, thenR is a finite subdirect sum of strictly prime near-rings. It is also proved that, ifR is a regular and right duo near-ring that satisfies a.c.c. (or d.c.c.) on annihilating ideals of the form Ann (Q), whereQ is an ideal ofR, thenR is a finite direct sum of near-ringsR _i (1 i n) where eachR _i is simple and strictly prime.     

Characterizingω 1 and the long line by their topological elementary reflections

Given a topological space 〈X, T〉 ∈M, an elementary submodel of set theory, we defineX _Mto beX ∩M with the topology generated by {U ∩M : U ∈T ∩M}. We prove that it is undecidable whetherX _Mhomeomorphic toω ₁ impliesX =X _M,yet it is true in ZFC that ifX _Mis homeomorphic to the long line, thenX =X _M.The former result generalizes to other cardinals of uncountable confinality while the latter generalizes to connected, locally compact, locally hereditarily LindelöfT ₂ spaces.     

Alcuni risultati sugli ideali massimali di anelli di polinomi

Riassunto In questo lavoro viene studiata la cardinalità minima dei sistemi di generatori di ideali massimaliM dell'anello di polinomiR[X ₁, …,X _n] (R dominio d'integrità) tali cheM⊃R=0. In particolare è dimostrato che sen≥2 edR è unS-dominio di dimensione 1, ogni siffatto ideale massimale può essere generato dan elementi.

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Summary This paper is concerned with estimates of the minimal number of generators for maximal idealsM in the polynomial ringR[X ₁, …,X _n] (R an integral domain) such thatM⊃R=0. In particular it is proved that ifn≥2 andR is a 1-dimensionalS-domain, every such maximal ideal can be generated byn elements.

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Lavoro eseguito nell'ambito dell'attività del G.N.S.A.G.A. del C.N.R.     

Annihilator Properties of Skew Monoid Rings

Let R be a ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u ₁,…, u _t } with 0 added, and M a factor of F setting certain monomial in U to 0, enough so that, for some n, M ⁿ  = 0. In this article, we study various annihilator properties and a variety of conditions and related properties that the skew monoid ring R[M; σ] is inherited from R.     

McCoy modules and related modules over commutative rings

Let M be a left R-module. Then M is a McCoy (resp., dual McCoy) module if for nonzero f(X)∈R[X] and m(X)∈M[X], f(X)m(X) = 0 implies there exists a nonzero r∈R (resp., m∈M) with rm(X) = 0 (resp., f(X)m = 0). We show that for R commutative every R-module is dual McCoy, but give an example of a non-McCoy module. A number of other results concerning (dual) McCoy modules as well as arithmetical, Gaussian, and Armendariz modules are given.     

Metrics on products of surfaces with non-positive sectional curvature

Let (S _i, g_i),i=1, 2 be two compact riemannian surfaces isometrically embedded in euclidean spaces. In this paper we show that ifM=S ₁×S₂,then for any functionF: M→R, the graph ofF, i.e. the manifold {(x, F(x)): x∈M}, does not have positive sectional curvature.     

Tychonoff products of compact spaces in ZF and closed ultrafilters

Let {(X_i, T_i): i ∈I } be a family of compact spaces and let X be their Tychonoff product. ??(X) denotes the family of all basic non‐trivial closed subsets of X and ??_R(X) denotes the family of all closed subsets H = V × ΠX_i of X, where V is a non‐trivial closed subset of ΠX_i and Q_H is a finite non‐empty subset of I. We show: (i) Every filterbase ?? ? ??_R(X) extends to a ??_R(X)‐ultrafilter ? if and only if every family H ? ??(X) with the finite intersection property (fip for abbreviation) extends to a maximal ??(X) family F with the fip. (ii) The proposition “if every filterbase ?? ? ??_R(X) extends to a ??_R(X)‐ultrafilter ?, then X is compact” is not provable in ZF. (iii) The statement “for every family {(X_i, T_i): i ∈ I } of compact spaces, every filterbase ?? ? ??_R(Y), Y = Π_{i ∈I}Y_i, extends to a ??_R(Y)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem. (iv) The statement “for every family {(X_i, T_i): i ∈ ω } of compact spaces, every countable filterbase ?? ? ??_R(X), X = Π_{i ∈ω}X_i, extends to a ??_R(X)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem restricted to countable families. (v) The countable Axiom of Choice is equivalent to the proposition “for every family {(X_i, T_i): i ∈ ω } of compact topological spaces, every countable family ?? ? ??(X) with the fip extends to a maximal ??(X) family ? with the fip” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

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Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).

     

A criterion for the primeness of ideals generated by polynomials with separated variables

Proving primeness of an idealI=〈f ₁, …,f _m〉 in a polynomial ringR=K[X ₁, …,X _n]ofn indeterminates over an algebraically closed fieldK is a difficult task in general. Although there are straightforward algorithms that decide whetherI is prime or not, they are prohibitively lengthy if the number of indeterminates or the degrees of thef _iare large. In this paper we will give an easy criterion for the primeness ofI if thef _iare polynomials with separated variables, i.e. no mixed monomials occur in thef _i. The work on this paper was done while the author was a MINERVA fellow at Tel Aviv University.     

Associated Prime Submodules of Finitely Generated Modules

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.     

Strict quasicomplements and the operators of dense imbedding

A quasicomplementM to a subspaceN of a Banach spaceX is called strict ifM does not contain an infinite-dimensional subspaceM ₁ such that the linear manifoldN+M ₁ is closed. It is proved that ifX is separable, thenN always admits a strict quasicomplement. We study the properties of the restrictions of the operators of dense imbedding to infinite-dimensional closed subspaces of a space where these operators are defined. Zaporozhye University, Zaporozhye. Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 46, No. 6, pp. 789–792, June, 1994     

Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF _σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.     

Principally Quasi-Baer skew Generalized Power Series modules

Let R be a ring and S a strictly totally ordered monoid. Let ω: S → End(R) be a monoid homomorphism. Let M _R be an ω-compatible module and either R satisfies the ascending chain conditions (ACC) on left annihilator ideals or every S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join. We show that M _R is p.q.-Baer if and only if the generalized power series module M[[S]] _{R[[S, ω]]} is p.q.-Baer. As a consequence, we deduce that for an ω-compatible ring R, the skew generalized power series ring R[[S, ω]] is right p.q.-Baer if and only if R is right p.q.-Baer and either R satisfies the ACC on left annihilator ideals or any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R. Examples to illustrate and delimit the theory are provided.     

Quasi-Armendariz Rings Relative to a Monoid

A ring R is called an M-quasi-Armendariz ring (a quasi-Armendariz ring relative to a monoid M) if whenever elements α = a ₁ g ₁ + a ₂ g ₂ + ··· + a _n g _n , β = b ₁ h ₁ + b ₂ h ₂ + ··· + b _m h _m  ? R[M] satisfy α R[M]β = 0, then a _i Rb _j  = 0 for each i, j. After discussing some basic properties of M-quasi-Armendariz rings, we consider the influence of transformation of the monoid M and the ring R on this property. Particularly, we give some sufficient conditions for the monoids M, N, and the ring R under which R is M × N-quasi-Armendariz if and only if R is M-quasi-Armendariz and N-quasi-Armendariz.     

Sequentially equidistant points in metric spaces

A sequence (z ₀,z ₁,z ₂,, ...,z _n, z _n+1) of points fromp=z ₀ toq=z _n+1 in a metric spaceX is said to besequentially equidistant ifd(z _i−1,z _i)=d(z _i,z _i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR ^m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results. Supported in part by NSF Grant DMS-8701666.