Let R be a ring and 𝒲 a self-orthogonal class of left R-modules which is closed under finite direct sums and direct summands. A complex C of left R-modules is called a 𝒲-complex if it is exact with each cycle Zn(C) ∈ 𝒲. The class of such complexes is denoted by 𝒞𝒲. A complex C is called completely 𝒲-resolved if there exists an exact sequence of complexes D· = … → D?1 → D0 → D1 → … with each term Di in 𝒞𝒲 such that C = ker(D0 → D1) and D· is both Hom(𝒞𝒲, ?) and Hom(?, 𝒞𝒲) exact. In this article, we show that C = … → C?1 → C0 → C1 → … is a completely 𝒲-resolved complex if and only if Cn is a completely 𝒲-resolved module for all n ∈ ?. Some known results are obtained as corollaries. 相似文献
Let ? be a prime ring, 𝒞 the extended centroid of ?, ? a Lie ideal of ?, F be a nonzero generalized skew derivation of ? with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) ? yx)n = 0 for all x, y ∈ ?, then ? is commutative and one of the following statements holds: (1) Either ? is central; (2) Or ? ? M2(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2. 相似文献
Let ? be a prime ring of characteristic different from 2, 𝒬r the right Martindale quotient ring of ?, 𝒞 the extended centroid of ?, F, G two generalized skew derivations of ?, and k ≥ 1 be a fixed integer. If [F(r), r]kr ? r[G(r), r]k = 0 for all r ∈ ?, then there exist a ∈ 𝒬r and λ ∈ 𝒞 such that F(x) = xa and G(x) = (a + λ)x, for all x ∈ ?. 相似文献
Consider an irreducible polynomial of the form f(X) = Xp ? aX ? b ∈ 𝔽[X] and α a root of f(X), where 𝔽 is a field of characteristic p. In 1975, F.J. Sullivan stated a lemma that provides the trace, taken with respect to the extension 𝔽(α)/𝔽, of elements of the form αn, where 0 ≤ n ≤ p2 ? 1. We present a generalization of Sullivan's Lemma and provide another proof of the original lemma. We explain how computing Tr(αn) for n < pr can be reduced to computing the traces Tr(αm) for all m ≤ r(p ? 1). 相似文献
Let (m, n) ∈ ℕ2, Ω an open bounded domain in ℝm, Y = [0, 1]m; uε in (L2(Ω))n which is two-scale converges to some u in (L2(Ω × Y))n. Let φ: Ω × ℝm × ℝn → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝm × ℝn, φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C1 > 0 and a function C2 ∈ L2(Ω) such that
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 E8) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups. 相似文献
In this paper we develop a time-independent approach for the study of the spectral shift function (SSF for short). We apply this method for the perturbed Stark Hamiltonian. We obtain a weak and a Weyl-type asymptotics with optimal remainder estimate of the SSF of the operator pair (P = P0 + V(x), P0 = ? h2Δ +x1), x = (x1,…, xn) where V(x) ∈ 𝒞∞(?n, ?) decays sufficiently fast at infinity, and h is a small positive parameter. Near a non-trapping energy λ, we give a pointwise asymptotic expansions in powers of h of the derivative of the SSF, and we compute explicitly the two leading terms. 相似文献
Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on Am such that (x1, …, xm) ≡ (y1, …, ym) if there exists a permutation σ on {1, …, m} such that yσ(i) = xi for all i. Let A(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A(m) are said to be adjacent if (x1, …, xm?1, a) ∈ X and (x1, …, xm?1, b) ∈ Y for some x1, …, xm?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A(m) to B(n) that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors. 相似文献
Let 𝒴 be a smooth connected manifold, Σ ? ? an open set and (σ, y) → 𝒫y(σ) a family of unbounded Fredholm operators D ? H1 → H2 of index 0 depending smoothly on (y, σ) ∈ 𝒴 × Σ and holomorphically on σ. We show how to associate to 𝒫, under mild hypotheses, a smooth vector bundle 𝒦 → 𝒴 whose fiber over a given y ∈ 𝒴 consists of classes, modulo holomorphic elements, of meromorphic elements φ with 𝒫yφ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators. 相似文献
Let 𝒜 be a unital prime ring containing a nontrivial idempotent P. Assume that Φ: 𝒜 → 𝒜 is a nonlinear surjective map. It is shown that Φ preserves strong commutativity if and only if Φ has the form Φ(A) = αA + f(A) for all A ∈ 𝒜, where α ∈ {1, ?1} and f is a map from 𝒜 into 𝒵(𝒜). As an application, a characterization of nonlinear surjective strong commutativity preserving maps on factor von Neumann algebras is obtained. 相似文献
Let 𝒜 = (An)n≥0 be an ascending chain of commutative rings with identity, S ? A0 a multiplicative set of A0, and let 𝒜[X] (respectively, 𝒜[[X]]) be the ring of polynomials (respectively, power series) with coefficient of degree i in Ai for each i ∈ ?. In this paper, we give necessary and sufficient conditions for the rings 𝒜[X] and 𝒜[[X]] to be S ? Noetherian. 相似文献
Denote by 𝒯n and 𝒮n the full transformation semigroup and the symmetric group on the set {1,…, n}, and ?n = {1} ∪ (𝒯n?𝒮n). Let 𝒯(X, 𝒫) denote the monoid of all transformations of the finite set X preserving a uniform partition 𝒫 of X into m subsets of size n, where m, n ≥ 2. We enumerate the idempotents of 𝒯(X, 𝒫), and describe the submonoid S = ? E ? generated by the idempotents E = E(𝒯(X, 𝒫)). We show that S = S1 ∪ S2, where S1 is a direct product of m copies of ?n, and S2 is a wreath product of 𝒯n with 𝒯m?𝒮m. We calculate the rank and idempotent rank of S, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of ?n. 相似文献
Let d and n be positive integers with n ≥ d + 1 and 𝒫 ? ?d an integral cyclic polytope of dimension d with n vertices, and let K[𝒫] = K[?≥0𝒜𝒫] denote its associated semigroup K-algebra, where 𝒜𝒫 = {(1, α) ∈ ?d+1: α ∈ 𝒫} ∩ ?d+1 and K is a field. In the present paper, we consider the problem when K[𝒫] is Cohen–Macaulay by discussing Serre's condition (R1), and we give a complete characterization when K[𝒫] is Gorenstein. Moreover, we study the normality of the other semigroup K-algebra K[Q] arising from an integral cyclic polytope, where Q is a semigroup generated by its vertices only. 相似文献
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(ag, b) = 1, under what conditions does the group G belong to 𝔚? In particular, we consider the n-Engel word w(x, y) = [x, ny]. 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
Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group 𝒪(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is 𝒪(V, I) = 𝒪(n). There is a nice result that 𝒪(n) ? Aut(𝔬(n)) over ? or ?, where 𝔬(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is “Yes” if it is GF(2). We show it explicitly with the combinatorial basis ?. This is a verification of Steinberg's main result in 1961, that is, Aut(𝔬(n)) is simple over the square field, with a nonsimple exception Aut(𝔬(5)) ? 𝒪(5) ? 𝔖6. 相似文献
Let L be a finite-dimensional complex simple Lie algebra, L? be the ?-span of a Chevalley basis of L, and LR = R ??L? be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A2, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A2, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism). 相似文献