Study of Hopficity in Certain Classes of Rings

The main results proved in this paper are:

1. For any non-zero vector space V _Dover a division ring D, the ring R= End(V _D) is hopfian as a ring

2. Let Rbe a reduced π-regular ring &; B(R) the boolean ring of idempotents of R. If B(R) is hopfian so is R.The converse is not true even when Ris strongly regular.

3. Let Xbe a completely regular spaceC(X) (resp. C ^?(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let Rbe any one of C(X) or C ^?(X). Then Ris an exchange ring if &; only if Xis zero dimensional in the sense of Katetov. for any infinite compact totally disconnected space X C(X) is an exchange ring which is not von Neumann regular.

4. Let Rbe a reduced commutative exchange ring. If Ris hopfian so is the polynomial ring R[T ₁,…,T _n] in ncommuting indeterminates over Rwhere nis any integer ≥ 1.

5. Let Rbe a reduced exchange ring. If Ris hopfian so is the polynomial ring R[T].     

Cohomology of general coinduced representions

ABSTRACT

Let R be a prime ring with a nonzero derivation d and let f(X ₁,…,X _t ) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x ₁,…,x _t )), f(x ₁,…,x _t )] _n  = 0 for all x _i  ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X ₁,…,X _t ) is centrally valued on R unless char R = 2 and dim _C RC = 4. We prove a more generalized version by replacing R with a left ideal.     

ADDITIVE SUBGROUPS GENERATED BY POLYNOMIAL VALUES ON RIGHT IDEALS

Let R be a prime ring with extended centroid C, ρ a nonzero right ideal of R and f(X ₁,…,X_t ) a nonzero polynomial over C. We determine the additive subgroup of RC generated by all elements f(x ₁,…,x_t ) for x ₁,…,x_t ∈ ρ C and obtain a result concerning an independence property of polynomials in prime rings.     

An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process

Let e _t=(e _t1,...e _tp) be a p-dimensional nonnegative strict white noise with finite second moments. Let h _ij(x) be nondecreasing functions from [0,) onto [0,) such that h _ij(x) x for i, j = 1,...,p. Let U = (u _ij) be a p×p matrix with nonnegative elements having all its roots inside the unit circle. Define a process X t=(X _t1,...,X _tp) by for

Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.     

Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

A note on the multiplicity in factor ring

Let (R,m) = k[x ₁,..., x _n ](x ₁,...,x _n ) be a local polynomial ring (k being an algebraically closed field), and Q:= (F ₁,..., F _r )R be a primary ideal in R with respect to a maximal ideal m ⊂ R. In this short note we give a formula for the multiplicity e ₀ (QR/(F ₁)R, R/(F ₁)R). The author was supported by the grant No. 1/0262/03) of the Slovak Ministry of Education.     

On the linearity of regression

Summary A stochastic process X={X _t :tT| is called spherically generated if for each random vector , there exist a random vector Y=(Y₁,..., Y _m) with a spherical (radially symmetric) distribution and a matrix A such that X is distributed as AY. X is said to have the linear regression property if (X ₀¦X ₁,..., X _n) is a linear function of X ₁,..., X _n whenever the X _j's are elements of the linear span of X. It is shown that providing the linear span of X has dimension larger than two, then X has the linear regression property if and only if it is spherically generated. The class of symmetric stable processes which are spherically generated is shown to coincide with the class of socalled sub-Gaussian processes, characterizing those stable processes having the linear regression property.This research was supported by a grant from the University of Wisconsin-Milwaukee     

Right ideals generated by an idempotent of finite rank

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let f(X₁,…,X_t) be an arbitrary and fixed polynomial over K in noncommuting indeterminates X₁,…,X_t with constant term 0 such that for some μK occurring in the coefficients of f(X₁,…,X_t). It is proved that a right ideal ρ of R is generated by an idempotent of finite rank if and only if the rank of f(x₁,…,x_t) is bounded above by a same natural number for all x₁,…,x_tρ. In this case, the rank of the idempotent that generates ρ is also explicitly given. The results are then applied to considering the triangularization of ρ and the irreducibility of f(ρ), where f(ρ) denotes the additive subgroup of R generated by the elements f(x₁,…,x_t) for x₁,…,x_tρ.     

Special isomorphisms of F[x 1,..., x n ] preserving GCD and their use

On the ring R = F[x ₁,..., x _n ] of polynomials in n variables over a field F special isomorphisms A’s of R into R are defined which preserve the greatest common divisor of two polynomials. The ring R is extended to the ring S: = F[[x ₁,..., x _n ]]⁺ and the ring T: = F[[x ₁,..., x _n ]] of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms A’s are extended to automorphisms B’s of the ring S. Using the property that the isomorphisms A’s preserve GCD it is shown that any pair of generalized polynomials from S has the greatest common divisor and the automorphisms B’s preserve GCD. On the basis of this Theorem it is proved that any pair of generalized polynomials from the ring T = F[[x ₁,..., x _n ]] has a greatest common divisor.     

Functions of elementary random variables and their application to system reliability

Summary LetX ₁,...,X _n be elementary random variables, i.e. random variables taking only finitely many values in . Then for an arbitray functionf(X ₁,...,X _n ) inX ₁,...,X _n a unique polynomial representation with the aid of Lagrange polynomials is given. This easily yields the moments as well as the distribution off(X ₁,...,X _n ) by terms of finitely many moments of the variablesX ₁,...,X _n . For n=1 a necessary and sufficient condition results thatm numbers are the firstm moments of a random variable takingm+1 different values. As an application of random functionsf(X ₁,...,X _n ) the reliability of technical systems with three states is treated.

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Zusammenfassung X ₁, ...,X _n seien elementare Zufallsvariable, d. h., Zufallsvariable, die nur endlich viele reelle Werte annehmen. Mit Hilfe von Lagrangepolynomen wird für eine beliebige Funktionf(X₁,...,X _n ) eine eindeutige polynomiale Darstellung angegeben. Daraus ergeben sich leicht die Momente wie auch die Verteilung von f(X₁,...,X _n ), ausgedrückt durch die Momente der VariablenX ₁,...,X _n . Fürn=1 erhält man eine notwendige und hinreichende Bedingung, daßm Zahlen die erstenm Momente einer Zufallsvariablen sind, diem+1 verschiedene Werte annimmt. Als Anwendung wird die Zuverlässigkeit eines technischen Systems mit drei Zuständen behandelt.

     

Toward Homological Characterization of Semirings: Serre’s Conjecture and Bass’s Perfectness in a Semiring Context

Among other results on homological characterization of semirings, we prove that the classes of projective and free right (left) semimodules over the polynomial semiring R[x₁, x₂,..., x_n] over an additively regular division semiring R coincide iff R is a field. Also it is shown that an additively regular commutative semiring R is perfect (in H. Basss sense) iff R is a perfect ring.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived July 27, 2003; accepted in final form April 2, 2004.     

Kinematical conservation laws in a space of arbitrary dimensions

In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. In this paper, we present the theory of kinematical conservation laws (KCL) in a space of arbitrary dimensions, i.e., d-D KCL, which are equations of evolution of a moving surface Ω_t in d-dimensional x-space, where x = (x ₁, x ₂,..., x _d) ∈ R^d. The KCL are derived in a specially defined ray coordinates (ξ = (ξ₁, ξ₂,..., ξ_d?1), t), where ξ₁, ξ₂,..., ξ_d?1 are surface coordinates on Ω_t and t is time. KCL are the most general equations in conservation form, governing the evolution of Ω_t with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ω_t when Ω_t is a curve in R² and is a curve on Ω_t when it is a surface in R³. Across a kink the normal n to Ω_t and normal velocity m on Ω_t are discontinuous.     

Folgenkonvexe funktionen

In this paper functions f:(a,b)R are considered with the property that for all n>2 and all x₁,x₂,...,x_n(a,b) is convex in k. Functions with this property are called sequentially convex. It is proved that if f is convex, twice differentiable, and f is convex then f is sequentially convex. In case f is a continous function defined on the whole ofR these conditions are necessary too.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

ALMOST COHEN–MACAULAY MODULES

A commutative noetherian ring R is called an almost Cohen–Macaulay ring if depth(P, R) = depth(P R _P , R _P ) for every P ∈ SpecR. [It was called a D-ring by Han (Acta Math. Sinica 1998, 4, 1047–1052).] Several fundamental properties of almost Cohen–Macaulay rings were estabished by Han. In this note, a new characterization is proved: R is an almost Cohen–Macaulay ring if and only if height P ≤ 1 + depth(P, R) for every P ∈ Spec(R). By this characterization, we settle an unsolved problem in Han's paper: R is an almost Cohen–Macaulay ring if and only if so is the power series ring R[[X ₁, …, X _n ]]. The notion of an almost Cohen–Macaulay ring is generalized to that of an almost Cohen–Macaulay module in this note.     

Ratio comparisons of supremum and stop rule expectations

Summary Suppose X ₁,X ₂,...,X_n are independent non-negative random variables with finite positive expectations. Let T _n denote the stop rules for X ₁,...,X _n. The main result of this paper is that E(max{X ₁,...,X _n }) <2 sup{EX _t :tT _n }. The proof given is constructive, and sharpens the corresponding weak inequalities of Krengel and Sucheston and of Garling.Partially supported by AFOSR Grant F49620-79-C-0123     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

Hypothesis testing for the common mean of two normal distributions in the presence of an indifference zone

Summary LetX ₁,...,X _m andY _t,...,Y be independent, random samples from populations which are N(θ,σ _x ² ) and N(θ,σ _y ² ), respectively, with all parameters unknown. In testingH ₀:θ=0 againstH ₁:θ≠0, thet-test based upon either sample is known to be admissible in the two-sample setting. If, however, one testsH ₀ againstH ₁:|θ|≧ε>0, with ε arbitrary, our main results show: (i) the construction of a test which is better than the particulart-test chosen, (ii) eacht-test is admissible under the invariance principle with respect to the group of scale changes, and (iii) there does not exist a test which simultaneously is better than botht-tests.     

Limit distributions of conditionalU-statistics

Let{(X_n, Y_n)}_n1 be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions ofU _n ^h (t), the so-calledconditional U-statistics, introduced by Stute.⁽¹⁰⁾ They are estimators of functions of the formm ^h (t)=E{h(Y ₁,...,Y _k )|X ₁=t ₁,...,X _k =t _k },t=(t ₁,...,t _k ) ^k whereE |h|<. Heret is fixed. In caset ₁=...=t_k=t (say), we describe the limiting random variables asmultiple Wiener integrals with respect toP _t, the conditional distribution ofY, givenX=t. Whent _i, 1ik, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.Research supported by National Board for Higher Mathematics, Bombay, India.