共查询到20条相似文献,搜索用时 31 毫秒
1.
《代数通讯》2013,41(9):2865-2877
ABSTRACT Let K denote a commutative ring with unity and A be a K-algebra. An element, d ∈ A is said to be left self distributive, or LSD, if dxy = dx dy for all x, y ∈ A. Let ?(A) be the set of LSD elements. Similarly, one can define the set of right self distributive, or RSD, elements and let ?(A) be the set of RSD elements. Let 𝒟(A) = ?(A) ∩ ?(A), the set of self distributive, or SD, elements. An algebra, A, is said to be left self distributively generated, or LSD-generated, if A = mod K (?(A)), the K-module generated by ?(A). Analogously, one defines RSD-generated and SD-generated algebras. If A = mod K (?(A)) = mod K (?(A)), then A is said to be LSD/RSD-generated, which is a strictly larger class than the class of SD-generated algebras. Examples are given to illustrate the variety of LSD-generated algebras. This paper continues the study of LSD-generated, RSD-generated, LSD/RSD-generated and SD-generated algebras. This paper characterizes exactly which structural matrix rings are LSD-generated. The paper begins with an important lemma that characterizes LSD elements in a matrix ring in terms of the entries of the matrix. The main result characterizes those structural matrix rings that are LSD-generated, first in terms of a 2 × 2 generalized matrix ring, then strictly in terms of the shape of the matrix ring. Sharper results are obtained for LSD/RSD-generated and SD-generated structural matrix rings. The final section is devoted to an application of this result to endomorphism rings. If the endomorphism ring of a finitely generated module is a homomorphic image of a structural matrix ring, then the module is a direct sum of cyclic modules. Further conditions are given to describe when the structural matrix ring is LSD-generated, in terms of the annihilators of the generating set. 相似文献
2.
Let A be a torsion-free abelian group and F a free subgroup of A. We prove that if A/F is a reduced p-group and A/(F + C) is reduced for every p-pure subgroup C of A, then A is free. Let KG be the group algebra of an abelian group G over a field K of prime characteristic p. Denote by S(KG) the p-component of the group V(KG) of normalized units of KG (of augmentation 1). Let H be an arbitrary group and KH ? KG as K-algebras. We prove the following. First, assume that G is a splitting group, the p-component G p of G is simply presented, and the field K is perfect. Then H p ? G p . If, in addition, G is p-mixed, then G p is a direct factor of S(KG), and G is a direct factor of V(KG), each with the same simply presented complement. Secondly, we introduce a class of special p-mixed abelian groups and prove that, if G belong to this class, then any group basis of the group algebra KG splits. Besides, H is p-mixed and splits. Thirdly, if G is a special p-mixed abelian group and G p is a reduced totally projective p-group, then H ? G. These results correct some essential inaccuracies and incompleteness in the proofs of results in this direction of Danchev [3-8]. 相似文献
3.
Let G be a finite group and cs(G) be the set of conjugacy class sizes of G. In 1987, J. G. Thompson conjectured that, if G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying that cs(G) = cs(M), then G ? M. This conjecture has been proved for Suzuki groups in [5]. In this article, we improve this result by proving that, if G is a finite group such that cs(G) = cs(Sz(q)), for q = 22m+1, then G ? Sz(q) × A, where A is abelian. We avoid using classification of finite simple groups in our proofs. 相似文献
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6.
《代数通讯》2013,41(6):2771-2789
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
7.
Let G be a finite group and δ(G) denote the number of conjugacy classes of all non-cyclic subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In [7], Meng and Li showed the inequality δ(G)≥2|π(G)|?2, where G is non-cyclic solvable group. In this paper, we describe the finite groups G such that δ(G) = 2|π(G)|?2. Another aim of this paper would show δ(G)≥M(G)+2 for unsolvable groups G and the equality holds ?G?A5 or SL(2,5), where M(G) denotes the number of conjugacy classes of all maximal subgroups of G. 相似文献
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9.
《代数通讯》2013,41(7):3529-3546
Abstract For an ideal I of a Noetherian local ring (R, m ) we consider properties of I and its powers as reflected in the fiber cone F(I) of I. In particular,we examine behavior of the fiber cone under homomorphic image R → R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψ J : F R (I) → F R′(IR′). We consider the structure of fiber cones F(I) for which ker ψ J ≠ 0 for each nonzero ideal J of R. If dim F(I) = d > 0,μ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence,we prove that if grade(G +(I)) ≥ d ? 1,then F(I) is Cohen-Macaulay and thus a hypersurface. 相似文献
10.
M. I. Elashiry 《代数通讯》2013,41(6):2132-2138
For any integer n ≥ 2, a group G is said to have the n-rewritable property R n if every infinite subset X of G contains n elements x 1,…, x n such that the product x 1…x n = x σ(1)…x σ(n) for some permutation σ ≠ 1. We show here that if G satisfies R n , then G has a subgroup N of finite index with a finite central subgroup A of N such that the exponent of (N/A)/Z(N/A) is finite and has size bounded by (n ? 1)!. This extends the main result in [4] which asserts that a group G is an R n group for some integer n if and only if G has a normal subgroup F such that G/F is finite, F is an FC-group, and the exponent of F/Z(F) is finite. 相似文献
11.
Martino Garonzi 《代数通讯》2013,41(2):471-491
12.
《代数通讯》2013,41(5):2381-2401
Abstract Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U n (R) be the subgroup of GL(V) that preserves ( , ). Let SU n (R) be the subgroup of all g ∈ U n (R) whose determinant is equal to one. Let Ψ be the Weil character of U n (R). All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU n (R). 相似文献
13.
We consider the distance enumerator Δ G (x) of a finite permutation group G, which is the polynomial ∑ g∈G x n?π(g), where n is the degree of G and π(g) the number of fixed points of g ∈ G. In particular, we introduce a bivariate polynomial which is a special case of the cycle index of G, and from which Δ G (x) can be obtained, and then use this new polynomial to prove some identities relating the distance enumerators of groups G and H with those of their direct and wreath products. In the case of the direct product, this answers a question of Blake et al. (1979). We also use the identity for the wreath product to find an explicit combinatorial expression for the distance enumerators of the generalised hyperoctahedral groups C m ? S n . 相似文献
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Let A be a regular multiplier Hopf algebra, and let Aut(A) denote the set of all isomorphisms α from A to itself that are algebra maps satisfying (Δ ○ α)(a) = (α ? α) ○ Δ(a) for all a ∈ A. Let G be a certain crossed product group Aut(A) × Aut(A). The main purpose of this article is to provide a class of new braided T-categories in the sense of Turaev [\citealp9]. For this, we introduce a class of new categories A 𝒴𝒟 A (α, β) of (α, β)-Yetter–Drinfel'd modules with α, β ∈Aut(A), and we show that the category ?𝒴𝒟(A) = { A 𝒴𝒟 A (α, β)}(α, β)∈G becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic [6]. 相似文献
15.
《代数通讯》2013,41(11):5627-5651
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Pavel Shumyatsky 《Monatshefte für Mathematik》2012,168(1):113-124
Let A be a group isomorphic with either S 4, the symmetric group on four symbols, or D 8, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C G (V)?=?1 and C G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms. 相似文献
17.
《代数通讯》2013,41(9):4195-4214
Abstract For a ring S, let K 0(FGFl(S)) and K 0(FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring Ris semiperfect if and only if the group homomorphism K 0(FGFl(R)) → K 0(FGFl(R/J(R))) is an epimorphism and K 0(FGFl(R)) = K 0(FGPr(R)). 相似文献
18.
Let A and B be multiplier Hopf algebras, and let R ∈ M(B ? A) be an anti-copairing multiplier, i.e, the inverse of R is a skew-copairing multiplier in the sense of Delvaux [5]. Then one can construct a twisted tensor coproduct multiplier Hopf algebra A ? R B. Using this, we establish the correspondence between the existence of quasitriangular structures in A ? R B and the existence of such structures in the factors A and B. We illustrate our theory with a profusion of examples which cannot be obtained by using classical Hopf algebras. Also, we study the class of minimal quasitriangular multiplier Hopf algebras and show that every minimal quasitriangular Hopf algebra is a quotient of a Drinfel’d double for some algebraic quantum group. 相似文献
19.
Let H, A and B be subgroups of a group G. We call the pair (A, B) a θ-pair for H in G if: (i) \({\langle H, A\rangle=G}\) and B = (A ∩ H) G ; (ii) if A 1/B is a proper subgroup of A/B and \({{A_1/B \vartriangleleft G/B}}\), then \({G\neq \langle H, A_1\rangle}\). In this paper, we study the θ-pairs for 2-maximal subgroups of a group, which imply a group to be solvable or supersolvable. 相似文献
20.
In [1], the authors computed the additive and multiplicative structure of HH*(A;A), where A is the n-th exterior algebra over a field. In this paper, we derive all their results using a different method (AMT) as well as calculate the additive structure of HHk(A;A) and HHk(A;A) over ?. We provide concise presentations of algebras HH?(A;A) and HH*(A;A) as well as determine their generators in the Hochschild complex. Finally, we compute an explicit free resolution (spanned by multisets) of the Ae-module A and describe the homotopy equivalence to its bar resolution. 相似文献