共查询到20条相似文献,搜索用时 15 毫秒
1.
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. 相似文献
2.
Abstract Let A be a commutative ring with identity, let X, Y be indeterminates and let F( X, Y), G( X, Y) ∈ A[ X, Y] be homogeneous. Then the pair F( X, Y), G( X, Y) is said to be radical preserving with respect to A if Rad(( F( x, y), G( x, y)) R) = Rad(( x, y) R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht( Q/ P) = 2. 相似文献
3.
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
4.
Abstract Given a contravariant functor F : 𝒞 → 𝒮 ets 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) : ? → 𝒮 ets, 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
5.
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. 相似文献
6.
Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ?( R) and ?( R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: -
If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ?(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/?(R t G) ? R t 1 (G/G p ) as R-algebras; -
If R is a ring of prime characterisitic p and R* is p-divisible, then ?(R t G) = 0 if and only if ?(R) = 0 and G p = 1; and -
If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ? R t 1 H 0 as R-algebras. 相似文献
7.
ABSTRACT A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ? (p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M m ( R) over an n-clean ring is n-clean, and the m× m ( m>1) matrix ring M m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a 1 R + ˙˙˙ a m R = dR, with m ≥ 2, a 1,…, a m, d ∈ R, there exist u 1 ∈ U(R) and u 2,…, u m ∈ W(R) such that a 1 u 1 + ? a m u m = Rd. 相似文献
8.
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}. 相似文献
9.
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
10.
Abstract A ring R is called strongly stable if whenever aR + bR = R, there exists a w ∈ Q( R) such that a + bw ∈ U( R), where Q( R) = { x ∈ R ∣ ? e ? e 2 ∈ J( R), u ∈ U( R) such that x = eu}. These rings are shown to be a natural generalization of semilocal rings and unit regular rings. We investigate the extensions of strongly stable rings. K 1-groups of such rings are also studied. In this way we recover and extend some results of Menal and Moncasi. 相似文献
11.
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
12.
Let ( R, m) be a Cohen–Macaulay local ring, and let ? = { F i } i∈? be an F 1-good filtration of ideals in R. If F 1 is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let ( R, m) be a Gorenstein local ring, and let F 1 be an ideal with ht( F 1) = g > 0. If there exists a reduction J of ? with μ( J) = g and reduction number u: = r J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J n : F u = F n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω G(?) is at most g and that of the canonical module ω R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J u : F u = F u . We illustrate with various examples cases where G(?) is or is not Gorenstein. 相似文献
13.
Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n( u, w) ≥1 such that ( F( uw) ? bwu) n = 0, then one of the following statements holds: F = 0 and b = 0; R ? M2(K), the ring of 2 × 2 matrices over a field K, b2 = 0, and F(x) = ?bx, for all x ∈ R. 相似文献
14.
ABSTRACT Let X = Spec( R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec( R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R 𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭). Then there exists a bijection γ 𝔪: Proj( G 𝔭( A) ? A/𝔭 k) → Proj( G( A 𝔭) ? k(𝔭) K) such that for every subset Σ of Proj( G 𝔭( A) ? A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ 𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety. 相似文献
16.
Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I 2) implies a ∈ I or b ∈ I. Let φ: ?( R) → ?( R) ∪ {?} be a function where ?( R) is the set of ideals of R. We call a proper ideal I of R a φ- prime ideal if a, b ∈ R with ab ∈ I ? φ( I) implies a ∈ I or b ∈ I. So taking φ ?( J) = ? (resp., φ 0( J) = 0, φ 2( J) = J 2), a φ ?-prime ideal (resp., φ 0-prime ideal, φ 2-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals. 相似文献
17.
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
18.
Suppose G 1 ? G are complex linear simple Lie groups. Let 1 ? be the corresponding pair of Lie algebras. For the Killing-orthogonal of 1 in we have a vector space direct sum = 1⊕ , which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G 1) = ( SL (4,?),Sp (2,?)); here = (4,?), 1 = (2,?), and = { X ? | X ? = X}, where X? X ? is the symplectic involution. We prove that G = G 1exp ∪ i G 1exp . The key point of the proof is to study in detail the set exp ; and for that purpose we introduce the J-twisted Pfaffian of size 2 n defined on the set of all 2 n × 2 n matrices X satisfying X ? = X, which is here a natural counterpart of the standard Pfaffian. 相似文献
19.
Let G be a finite group. A subgroup K of a group G is called an ?- subgroup of G if N G ( K) ∩ K x ≦ K for all x ? G. The set of all ?-subgroups of G will be denoted by ?( G). Let P be a nontrivial p-group. A chain of subgroups 1 = P 0 ? P 1 ? ··· ? P n = P is called a maximal chain of P provided that | P i : P i?1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P 0 ? P 1 ? ··· ? P i ? ··· ? P n = P such that P i ? ?( G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups. 相似文献
20.
We show that if R is an infinite ring such that XY ∩ YX ≠ ? for all infinite subsets X and Y, then R is commutative. We also prove that in an infinite ring R, an element a ∈ R is central if and only if aX ∩ Xa ≠ ? for all infinite subsets X. 相似文献
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