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1.
Nik Stopar 《代数通讯》2013,41(6):2053-2065
We describe surjective additive maps θ: A → B which preserve zero products, where A is a ring with a nontrivial idempotent and B is a prime ring. We also characterize surjective additive maps θ: A → B such that for all x, y ∈ A we have θ(x)θ(y)* = 0 if and only if xy* = 0. Here A is a unital prime ring with involution that contains a nontrivial idempotent and B is a prime ring with involution.  相似文献   

2.
O. Macedońska 《代数通讯》2013,41(12):4661-4667
Let F = ?x, y? be a free group. It is known that the commutator [x, y ?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ?1] ≡ u, where u ∈ [F′, ?x?].  相似文献   

3.
Gordon A. Swain 《代数通讯》2013,41(5):1613-1620
For a prime ring A with involution, we explore the characterization of additive bijective maps φ: A → A such that φ(x)φ(y)* = 0 whenever xy* = 0. In particular, we show that if A is prime, unital, and generated by nontrivial idempotents, then there is a *-monomorphism g of A into Q, and an element r ∈ A such that φ(x) = rg(x) for all x ∈ A.  相似文献   

4.
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.  相似文献   

5.
Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x1,…, xn) 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,…, rn) ∈ 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,…, xn)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) = ax + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x1,…, xn)2 is central valued on R.

  相似文献   

6.
Asma Ali  Faiza Shujat 《代数通讯》2013,41(9):3699-3707
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.

  相似文献   

7.
《代数通讯》2013,41(5):2053-2065
Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.  相似文献   

8.
《代数通讯》2013,41(2):869-875
Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair (X, x) where X is an object of 𝒞 and x ∈ F(X) if for any object Y of 𝒞 and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ?xt (?, M) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.  相似文献   

9.
For a triangular algebra 𝒜 and an automorphism σ of 𝒜, we describe linear maps F,G:𝒜𝒜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y𝒜 are such that xy = 0. In particular, when 𝒜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x𝒜, where D:𝒜𝒜 is a skew derivation. When 𝒜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G.  相似文献   

10.
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.

  相似文献   

11.
Timothy J. Ford 《代数通讯》2013,41(9):3277-3298
We study algebra classes and divisor classes on a normal affine surface of the form z 2 = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z 2 ? f), and if R = k[x, y][f ?1] and S = R[z]/(z 2 ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H1(G, Cl(T)) → H1(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H1(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples.  相似文献   

12.
Willian Franca 《代数通讯》2018,46(7):2890-2898
Let R be a unital simple ring. Under some technical restrictions, we characterize m-linear mappings G:RmR satisfying [G(u,…,u),u]?=?0 for all unit uR.  相似文献   

13.
AA-Rings     
《代数通讯》2013,41(10):3853-3860
Abstract

Let R be a ring with identity such that R +, the additive group of R, is torsion-free of finite rank (tffr). The ring R is called an E-ring if End(R +) = {x ? ax : a ∈ R} and is called an A-ring if Aut(R +) = {x ? ux : u ∈ U(R)}, where U(R) is the group of units of R. While E-rings have been studied for decades, the notion of A-rings was introduced only recently. We now introduce a weaker notion. The ring R, 1 ∈ R, is called an AA-ring if for each α ∈ Aut(R +) there is some natural number n such that α n  ∈ {x ? ux : u ∈ U(R)}. We will find all tffr AA-rings with nilradical N(R) ≠ {0} and show that all tffr AA-rings with N(R) = {0} are actually E-rings. As a consequence of our results on AA-rings, we are able to prove that all tffr A-rings are indeed E-rings.  相似文献   

14.
A weak Cayley table isomorphism is a bijection φ:GH of groups such that φ(xy)~φ(x)φ(y) for all x,yG. Here ~ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ:GG forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I:GG,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.  相似文献   

15.
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.  相似文献   

16.
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.  相似文献   

17.
18.
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.  相似文献   

19.
Cihat Abdioğlu 《代数通讯》2017,45(4):1741-1756
Let R be a noncommutative prime ring with extended centroid C and maximal left ring of quotients Qml(R). The aim of the paper is to study a basic functional identity concerning bi-additive maps on R. Precisely, it is proved that a bi-additive map B:R×RQml(R) satisfying [B(x,y),[x,y]] = 0 for all x,yR must be of the form (x,y)?λ[x,y]+μ(x,y) for x,yR, where λ∈C and μ:R×RC is a bi-additive map. As applications to the theorem, Jordan σ-biderivations with σ an epimorphism and additive commuting maps on noncommutative Lie ideals of R are characterized.  相似文献   

20.
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.  相似文献   

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