Minimal differential identities in prime rings

It is known that a prime ring which satisfies a polynomial identity with derivations applied to the variables must satisfy a generalized polynomial identity, but not necessarily a polynomial identity. In this paper we determine the minimal identity with derivations which can be satisfied by a non-PI prime ringR. The main result shows, essentially, that this identity is the standard identityS ₃ withD applied to each variable, whereD = ad(y) fory inR, y ² = 0, andy of rank one in the central closure ofR.     

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

We studied the solvability of the algebra which satisfies the polynomial identity (x ²)² = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x ²)² = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim  _F A ≤ 7.     

Unitary Units Satisfying a Group Identity

Let K be a nonabsolute field of characteristic p ≠ 2, G a locally finite group and KG its group algebra. Let ?: KG → KG denote the K-linear extension of an involution ? defined on G. In this article, we prove that if the subgroup 𝒰_?(KG), i.e., the ?-unitary units of KG, satisfies a group identity, then KG satisfies a polynomial identity. Moreover, in case the prime radical of KG is nilpotent, we characterize the groups G for which 𝒰_?(KG) satisfies a group identity.     

ON NEAR-RINGS WITH MATRIX UNITS

Abstract

In ring theory it is well known that a ring R with identity is isomorphic to a matrix ring if and only if R has a set of matrix units. In this paper, the above result is extended to matrix near-rings and it is proved that a near-ring R with identity is isomorphic to a matrix near-ring if and only if R has a set of matrix units and satisfies two other conditions. As a consequence of this result several examples of matrix near-rings are given and for a finite group (Γ, +) with o(Γ) > 2 it is proved that M₀ (Γⁿ) is (isomorphic to) a matrix near-ring.     

Skew Derivations and Engel Conditions

It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x ^{k ₀} ), x ^{k ₁} ], x ^{k ₂} ],…], x ^{k _n} ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result in two variables is obtained for d a (θ, ?)-derivation.     

Lattice Points on Similar Figures and Conics

Let us say that a plane figure F satisfies Steinhaus’ condition if for any positive integer n, there exists a figure F _n similar to F which satisfies the condition |F_n?\mathbb Z²|=n{|F_n\cap{\mathbb Z}^2|=n}. For example, the circular disc satisfies Steinhaus’ condition. We prove that every compact convex region in the plane \mathbb R²{\mathbb R^2} satisfies Steinhaus’ condition. As for plane curves, it is known that the circle satisfies Steinhaus’ condition. We consider Steinhaus’ condition for other conics, and present several results.     

A Generalization of Engel Conditions with Derivations in Rings

Let R be a prime ring of characteristic different from 2 with Z the center of R and d a nonzero derivation of R. Let k, m, n be fixed positive integers. If ([d(x ^k ), x ^k ] _n ) ^m  ∈ Z for all x ∈ R, then R satisfies S ₄, the standard identity in 4 variables.     

Derivations with power central values on Lie ideals in prime rings

Let R be a prime ring of char R ≠ 2 with a nonzero derivation d and let U be its noncentral Lie ideal. If for some fixed integers n ₁ ⩾ 0, n ₂ ⩾ 0, n ₃ ⩾ 0, (u ⁿ¹ [d(u), u]u ⁿ²) ⁿ³ ∈ Z(R) for all u ∈ U, then R satisfies S ₄, the standard identity in four variables.     

Centralizers satisfying polynomial identities

The following results are proved: IfR is a simple ring with unit, and for someaεR witha ⁿ in the center ofR, anyn, such that the centralizer ofa inR satisfies a polynomial identity of degreem, thenR satisfies the standard identity of degreenm. WhenR is not simple,R will satisfy a power of the same standard identity, provided thata andn are invertible inR. These theorems are then applied to show that ifG is a finite solvable group of automorphisms of a ringR, and the fixed points ofG inR satisfy a polynomial identity, thenR satisfies a polynomial identity, providedR has characteristic 0 or characteristicp wherep✗|G|. This research was supported in part by NSF Grant No. GP 29119X.     

Power-Centralizing Automorphisms of Lie Ideals in Prime Rings

Let R be a prime ring with center Z, L a noncentral Lie ideal of R, and σ a nontrivial automorphism of R such that [u ^σ,u] ⁿ  ∈ Z for all u ∈ L. If either char(R) > n or char(R) = 0, then R satisfies s ₄, the standard identity in 4 variables.     

A strange example on Property Np

 We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfies Property N ₁₀ , the 10-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M ² does not satisfy Property N ₁₀ . Thus this is a completely convincing example showing that surprisingly it is not true that if a line bundle M satisfies Property N _p then any power of M satisfies Property N _p . We recall that in [Ru] we proved the following statement: if M is a line bundle on a smooth complex projective variety and M satisfies Property N _p then M ^s satisfies Property N _p if s≥p. Received: 5 March 2001     

Identities in rings with involutions

A ringR with an involutiona →a* which satisfies a polynomial identityp[x ₁,…,x _d ;x*₁, …,x* _d ]=0 satisfies- an identity which does not include thex*. This generalizes the result of [1] where the symmetric elements ofR were assumed to satisfy an identity.     

Functional analytic aspects of non-commutative harmonic analysis

Given a homogeneous space X = G/H with an invariant measure it is shown, using Grothendieck's inequality, that a G-invariant Hilbert subspace of the space of distributions of order zero on X is actually contained in L_loc²(X). Moreover, if θ is an automorphism on G appropriately related to H, it is shown that, under condition that H-orbits are smooth, an H-bi-invariant distribution of positive type on G satisfies the identity Ť^θ = T if the corresponding Hilbert space is contained in L_loc²(X). This shows that, under the smooth orbit condition, G-invariant Hilbert subspaces of L_loc² (X) have a unique decomposition into irreducible Hilbert spaces as in the case of generalized Gelfand pairs.     

Algebres de rang 3 de composition faible

We study algebras of rank three admitting a nonzero quadratic form ?which satisfies the identity ?(x ²) = ?(x ²), and we show the relationship with power-associativity. By means of the Wedderburn decomposition relative to their radical, we prove that power-associative rank three algebras are actually Jordan.     

Generalized derivations and left multipliers on Lie ideals

Let m, n be two fixed positive integers and let R be a 2-torsion free prime ring, with Utumi quotient ring U and extended centroid C. We study the identity F(x ^m+n+1) = F(x)x ^m+n  + x ^m D(x)x ⁿ for x in a non-central Lie ideal of R, where both F and D are generalized derivations of R and then determine the relationship between the form of F and that of D. In particular the conclusions of the main theorem say that if D is the non-zero map in R, then R satisfies the standard identity s ₄(x ₁, . . . , x ₄) and D is a usual derivation of R.     

Symmetric units and group identities

In this paper we study rings R with an involution whose symmetric units satisfy a group identity. An important example is given by FG, the group algebra of a group G over a field F; in fact FG has a natural involution induced by setting g?g ⁻¹ for all group elements g∈G. In case of group algebras if F is infinite, charF≠ 2 and G is a torsion group we give a characterization by proving the following: the symmetric units satisfy a group identity if and only if either the group of units satisfies a group identity (and a characterization is known in this case) or char F=p >0 and 1) FG satisfies a polynomial identity, 2) the p-elements of G form a (normal) subgroup P of G and G/P is a Hamiltonian 2-group; 3) G is of bounded exponent 4p ^s for some s≥ 0. Received: 8 August 1997     

Minimizing the L ∞ Norm of the Gradient with an Energy Constraint

ABSTRACT

The variational problem in L ^∞ considered is to minimize F(u) = ‖Du‖ _{L ^∞(Ω)} subject to ∈ t _Ω |Du|² dx ≤ E for given E > 0. It is proven that a constrained minimizer exists and satisfies an Aronsson-Euler equation in the viscosity sense which depends on a parameter Λ_∞ ≥ 0. This parameter splits Ω into two parts. In one part the minimizer satisfies the infinity laplace equation and in the remaining part the minimizer is the solution of the elasto-plastic torsion problem with constraint ‖Du‖ _{L ^∞}  ≤ Λ_∞.     

Effective permeability of the boundary of a domain

Let u be a solution of the Laplace equation in a boilnded domain Ω in R ⁿ whose boundary consists of two disjoint surfaces λ₀ and λ₁. On λ₁ u satisfies a Dirichlet boundary condition, whereas on Λ₀ u satisfies the impermeability condition ?u/?n = 0 ever3 where except on m, small “holes” λ, separated from one another by distance of older of magnitude ε. and u= O on the holes. In this paper wr study the effective periwabilitv of the surface λ₀as ε→0.     

On the dualities between categories of k-completely regular modules

A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S at least two distinct words of length n on these letters are equal in S. Let U(A) denote the group of units of an associative algebra A over an infinite field of characteristic p > 0. We show that if A is unitally generated by its nilpotent elements then the following conditions are equivalent: U(A) is collapsing; U(A) satisfies some semigroup identity; U(A) satisfies an Engel identity; A satisfies an Engel identity when viewed as a Lie algebra; and, A satisfies a Morse identity. The characteristic zero analogue of this result was proved by the author in a previous paper.