Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r ^k ), r ^k ] _n  = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c ? α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

Remotal Sets in Orlicz Spaces

Let X be a Banach space, (I, μ) be a finite measure space. By L ^Φ(I, X), let us denote the space of all X-valued Bochner Orlicz integrable functions on the unit interval I equipped with the Luxemburg norm. A closed bounded subset G of X is called remotal if for any x ∈ X, there exists g ∈ G such that ‖x ? g‖ = ρ(x, G) = sup {‖x ? y‖: y ∈ G}. In this article, we show that for a separable remotal set G ? X, the set of Bochner integrable functions, L ^Φ(I, G) is remotal in L ^Φ(I, X). Some other results are presented.     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

On Certain Weak Engel-Type Conditions in Groups

Let w(x, y) be a word in two variables and 𝔚 the variety determined by w. In this paper we raise the following question: if for every pair of elements a, b in a group G there exists g ∈ G such that w(a ^g , b) = 1, under what conditions does the group G belong to 𝔚? In particular, we consider the n-Engel word w(x, y) = [x, _n y]. We show that in this case the property is satisfied when the group G is metabelian. If n = 2, then we extend this result to the class of all solvable groups.     

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x ₁,…, x _n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r ₁,…, r _n  ∈ R, a[F ²(f(r ₁,…, r _n )), f(r ₁,…, r _n )] = 0, then one of the following statements hold: 1. a = 0;

2. There exists λ ∈C such that F(x) = λx, for all x ∈ R;

3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c ² ∈ C;

4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c ² ∈ C.

     

Fundamental Sets of Functions on Locally Compact Abelian Groups

Let G be a locally compact Abelian group, and let X be a compact set of G. Given a positive definite function ?: G × G → ? whose real part is continuous at neutral element of G, we research a necessary and sufficient setting for the linear span of the set {x ∈ X → ?(x ? y): y ∈ X} to be dense in C(X) in the topology of uniform convergence. The context treated that is abstract encompasses classical cases of the literature, while other examples are entirely new.     

Weak Cayley table groups III: PSL(2,q)

A weak Cayley table isomorphism is a bijection φ:G→H of groups such that φ(xy)～φ(x)φ(y) for all x,y∈G. Here ～ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ:G→G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I:G→G,x?x^?1. Let 𝒲₀(G) = ?Aut(G),I?≤𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that PSL(2,pⁿ) has trivial weak Cayley table group, where p≥5 is a prime and n≥1.     

A Principle for Critical Point Under Generalized Regular Constraint and Ill-Posed Lagrange Multipliers Under Nonregular Constraints

In this article, a kind of nonregular constraint and a principle for seeking critical point under the constraint are presented, where no Lagrange multiplier is involved. Let E, F be two Banach spaces, g: E → F a c ¹ map defined on an open set U in E, and the constraint S = the preimage g ^?1(y ₀), y ₀ ∈ F. A main deference between the nonregular constraint and regular constraint is that g′(x) at any x ∈ S is not surjective. Recently, the critical point theory under the nonregular constraint is a concerned focus in optimization theory. The principle also suits the case of regular constraint. Coordinately, the generalized regular constraint is introduced and the critical point principle on generalized regular constraint is established. Let f: U → ? be a nonlinear functional. While the Lagrange multiplier L in classical critical point principle is considered, its expression is given by using generalized inverse g′⁺(x) of g′(x) as follows: if x ∈ S is a critical point of f| _S , then L = f′(x) ○ g′⁺(x) ∈ F*. Moreover, it is proved that if S is a regular constraint, then the Lagrange multiplier L is unique; otherwise, L is ill-posed. Hence, in case of the nonregular constraint, it is very difficult to solve Euler equations; however, it is often the case in optimization theory. So the principle here seems to be new and applicable. By the way, the following theorem is proved: if A ∈ B(E, F) is double split, then the set of all generalized inverses of A, GI(A) is smooth diffeomorphic to certain Banach space. This is a new and interesting result in generalized inverse analysis.     

Ascending Nets of Prüfer V-Multiplication Domains

Abstract

Let D be an integral domain, S ? D a multiplicative set such that aD _S  ∩ D is a principal ideal for each a ∈ D and let D ^(S) = ? _s∈S D[X/s]. It is known that if D is a Prüfer v-multiplication domain (resp., generalized GCD domain, GCD domain), then so is D ^(S) respectively. When D is a Noetherian domain, we obtain a similar result for the power series analog D ^((S)) = ? _s∈S D[[X/s]] of D ^(S). Our approach takes care simultaneously of both cases D ^(S) and D ^((S)).     

Group Rings Whose Symmetric Units Generate an n-Engel Group

Let F be an infinite field of characteristic different from 2 and G a torsion group. Write 𝒰⁺(FG) for the set of units in the group ring FG that are symmetric with respect to the classical involution induced from the map g ? g ^?1, for all g ∈ G. We classify the groups such that ?𝒰⁺(FG)? is n-Engel.     

A New Characterization of A 10 by its Noncommuting Graph

Let G be a nonabelian group and associate a noncommuting graph ?(G) with G as follows: The vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture.

Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ? M, where N(G):={n ∈ ? | G has a conjugacy class of size n}.

In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows.

AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ?(G) ? ? (M). Then G ? M.

In this short article we prove that if G is a finite group with ?(G) ? ? (A ₁₀), then G ? A ₁₀, where A ₁₀ is the alternating group of degree 10.     

Commuting Traces of Biadditive Maps on Invertible Elements

Let R be a simple unital ring. Under a mild technical restriction on R, we will characterize biadditive mappings G: R² → R satisfying G(u, u)u = uG(u, u), and G(1, r) = G(r, 1) = r for all unit u ∈ R and r ∈ R, respectively. As an application, we describe bijective linear maps θ: R → R satisfying θ(xyx^?1y^?1) = θ(x)θ(y)θ(x)^?1θ(y)^?1 for all invertible x, y ∈ R. This solves an open problem of Herstein on multiplicative commutators. More precisely, we will show that θ is an isomorphism. Furthermore, we shall see the existence of a unital simple ring R′ without nontrivial idempotents, that admits a bijective linear map f: R′ → R′, preserving multiplicative commutators, that is not an isomorphism.     

A General Theory of Almost Splitting Sets

Let * be a star-operation of finite type on an integral domain D. In this paper, we generalize and study the concept of almost splitting sets. We define a saturated multiplicative subset S of D to be an almost g*-splitting set of D if for each 0 ≠ d ∈ D, there exists an integer n = n(d) ≥1 such that d ⁿ  = st for some s ∈ S and t ∈ D with (t, s′)_* = D for all s′ ∈ S. Among other things, we prove that every saturated multiplicative subset of D is an almost g*-splitting set if and only if D is an almost weakly factorial domain (AWFD) with *-dim (D) = 1. We also give an example of an almost g*-splitting set which is not a g*-splitting set.     

Valuation and Pseudovaluation Subrings of an Integral Domain

Let R, S be two rings. We say that R is a valuation subring of S (R is a VD in S, for short) if R is a proper subring of S and whenever x ∈ S, we have x ∈ R or x ^?1 ∈ R. We denote by Nu(R) the set of all nonunit elements of a ring R. We say that R is a pseudovaluation subring of S (R is a PV in S, for short) if R is a proper subring of S and x ^?1 a ∈ R, for each x ∈ S?R, a ∈ Nu(R). This article deals with the study of valuation subrings and pseudovaluation subrings of a ring; interactions between the two notions are also given. Let R be a PV in S; the Krull dimension of the polynomial ring on n indetrminates over R is also computed.     

Commuting Values of Generalized Derivations on Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f(x ₁,…, x _n ) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f(R) the set of all evaluations of the polynomial f(x ₁,…, x _n ) in R. If [G(u)u, G(v)v] = 0, for any u, v ∈ f(R), we prove that there exists c ∈ U such that G(x) = cx, for all x ∈ R and one of the following holds: 1. f(x ₁,…, x _n )² is central valued on R;

2. R satisfies s ₄, the standard identity of degree 4.

     

Divisibility Graph for Symmetric and Alternating Groups

Let X be a nonempty set of positive integers and X* = X?{1}. The divisibility graph D(X) has X* as the vertex set, and there is an edge connecting a and b with a, b ∈ X* whenever a divides b or b divides a. Let X = cs(G) be the set of conjugacy class sizes of a group G. In this case, we denote D(cs(G)) by D(G). In this paper, we will find the number of connected components of D(G) where G is the symmetric group S _n or is the alternating group A _n .     

Supercentralizing Superderivations on Prime Superalgebras

ABSTRACT

Let A = A ₀ ⊕ A ₁ be an associative superalgebra over a commutative associative ring F, and let Z _s (A) be its supercenter. An F-mapping f of A into itself is called supercentralizing on a subset S of A if [x, f(x)] _s  ∈ Z _s (A) for all x ∈ S. In this article, we prove a version of Posner's theorem for supercentralizing superderivations on prime superalgebras.     

On Finite p-Groups Whose IA-Automorphisms Are All Class Preserving

Let G be a finite non-abelian p-group, where p is a prime. An automorphism α of G is called a class preserving automorphism if α(x) ∈ x ^G the conjugacy class of x in G, for all x ∈ G. An automorphism α of G is called an IA-automorphism if x ^?1α(x) ∈ G′ for each x ∈ G. In this paper, we give necessary and sufficient conditions on finite p-group G of nilpotency class 2 such that every IA-automorphism is class preserving.