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1.
M. Chacron 《代数通讯》2013,41(9):3325-3339
We are given a division ring D with involution (*) and with a *-valuation V such that V(aa*b ? baa*) >0 V(aa*b), for all nonzero elements a, b of D. We assume, further, that the associated residue class division ring D V is a (commutative) field with characteristic 0. In this work, we evidence two criteria in order for D to be either a standard quaternion division algebra or else a purely transcendental extension of its center. We apply one of these to answer our open Question 2.4.3 [2]. 相似文献
2.
M. Chacron 《代数通讯》2013,41(11):4613-4631
Let D be a division ring with centre Z and with involution (*). Let V be a valuation of D with value group Γ, a linearly ordered additive group (non necessarily commutative) together with a symbol ∞ (positive infinity). We assume that for each nonzero symmetric element s = s* of D, which is algebraic over Z, we have for all nonzero elements x of D, V(xa ? ax) > V(ax). We define the residue characteristic exponent p of V to be the characteristic χ of the associated residue division ring written as D V , if χ ≠ 0, and p = 1, if χ = 0. We show here that if F is a finite dimensional commutative subalgebra of D over Z, which is *-closed (i.e., F* = F), and if (*) is of the first kind (i.e., each central element of D must be symmetric), then [F: Z] = 2 r p m where m is a nonnegative integer and r = 0 or 1 according as the restricted involution in F is trivial or not. The case of an involution (*) of the second kind (i.e., some central element of D is not symmetric) requires (for this author) a stronger type of valuation, namely, V is a *-valuation, that is to say, for all elements x of D, we have V(x*) = V(x), a condition which readily implies Γ must be Abelian. Here, we can show that for F as in the preceding, [F: Z] = p m , where m is again a nonnegative integer. The preceding results generalize a theorem of Gräter and improve in parts recent theorems of this author in [2]. In the special case p = 2 the results provide a modicum of answers to the questions opened informally in [2] (see concluding paragraph in [2] or here Question 3.2.1). More is to be said in the third and final section of this work. 相似文献
3.
4.
We consider three infinite families of cyclic presentations of groups, depending on a finite set of integers and having the same polynomial. Then we prove that the corresponding groups with the same parameters are isomorphic, and that the groups are almost all infinite. Finally, we completely compute the maximal Abelian quotients of such groups, and show that their HNN extensions are high-dimensional knot groups. Our results contain as particular cases the main theorems obtained in two nice articles: Johnson et al. (1999) and Havas et al. (2001). 相似文献
5.
Huanyin Chen 《代数通讯》2013,41(4):1352-1362
An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8, Theorems 2.7 and 3.2], [9, Theorem 2.6], and [11, Theorem 7]. 相似文献
6.
Adolf Mader 《代数通讯》2013,41(8):2823-2844
The unique largest regular ideal Reg(A, A) in the endomorphism ring End(A) is computed for abelian groups A using the general tools developed in [7]. This generalizes earlier results on groups with regular endomorphism ring. Interesting questions remain for a very special class of mixed abelian groups. 相似文献
7.
Anna Giordano Bruno 《代数通讯》2013,41(11):4155-4174
For a set Γ, a function λ: Γ → Γ and a nontrivial abelian group K, the \emphgeneralized shift σλ: K Γ → K Γ is defined by (x i ) i∈Γ ? (x λ(i)) i∈Γ [3]. In this article we compute the algebraic entropy of σλ; it is either zero or infinite, depending exclusively on the properties of λ. This solves two problems posed in [2]. 相似文献
8.
Katsutoshi Amano 《代数通讯》2013,41(5):1811-1823
In a previous article (Amano and Masuoka, 2005), the author and Masuoka developed a Picard–Vessiot theory for module algebras over a cocommutative pointed smooth Hopf algebra D. By using the notion of Artinian simple (AS)D-module algebras, it generalizes and unifies the standard Picard–Vessiot theories for linear differential and difference equations. The purpose of this article is to define the notion of Liouville extensions of AS D-module algebras and to characterize the corresponding Picard–Vessiot group schemes. 相似文献
9.
This article is a sequel of [4], where we defined supervaluations on a commutative semiring R and studied a dominance relation ? ≥ ψ between supervaluations ? and ψ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation ?: R → U is a multiplicative map from R to a supertropical semiring U, cf. [4], [7], [8], [5], [9], with further properties, which mean that ? is a sort of refinement, or covering, of an m-valuation (= monoid valuation) v: R → M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [1], while ? ≥ ψ means that ψ: R → V is a sort of coarsening of the supervaluation ?. If ?(R) generates the semiring U, then ? ≥ ψ iff there exists a “transmission” α: U → V with ψ = α ○ ?. Transmissions are multiplicative maps with further properties, cf. [4, Section 5]. Every semiring homomorphism α: U → V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the article we study surjective transmissions via equivalence relations on supertropical semirings. We put special emphasis on homomorphic equivalence relations. Even those are often much more complicated than congruences by ideals in usual commutative algebra. 相似文献
10.
The article considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter example of Serrin [18], it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia [19] and Boccardo-Galouët [4] and the two notions of solutions of [2, 7], and show that they are equivalent. As side results, we reformulate the solution in the sense of [19], and prove the existence of solutions in the sense of [2, 4, 7] in case of mixed boundary conditions. 相似文献
11.
Yunchuan Yin 《代数通讯》2013,41(2):547-565
ABSTRACT The “W-graph” concept was introduced by Kazhdan and Lusztig in their influential article Kazhdan and Lusztig (1979). If W is a Coxeter group, then a W-graph provides a method for constructing a matrix representation of the Hecke algebra ? associated with W (the degree of the representation being the number of vertices of the W-graph). The aim of this note is to explicitly construct all the irreducible representations of ? when W is of type D 4 and D 5. 相似文献
12.
Jiangtao Shi 《代数通讯》2013,41(10):3916-3922
As an important application of Thompson's theorem [9, Theorem 10.4.2], a finite group is solvable if it has an abelian maximal subgroup. In this article, we mainly investigate the influence of some quantitative properties of abelian subgroups on solvability of finite groups. Some new results are obtained. 相似文献
13.
ABSTRACTLet I be a monomial ideal with minimal monomial generators m1,…, ms, and assume that deg(m1) ≥deg(m2) ≥ … ≥deg(ms). Among other things, we prove that the arithmetic degree of I is bounded above by deg(m1)…deg(mmht(I)), where mht(I) is the maximal height of associated primes of I. This bound is shaper than the one given in [12] and more natural than the one given in [9]. In addition, we point out that adeg(I) ≠ adeg(Gin(I)) in general and conjecture that adeg(I) = adeg(Gin(I)) if and only if R/I is sequentially Cohen–Macaulay. 相似文献
14.
Álvaro Muñoz 《代数通讯》2018,46(9):3873-3888
In this paper we give a complete classification of pointed fusion categories over ? of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules following the procedure presented in [9, 11]. The results of this paper permit to recover the classification of twisted quantum doubles of groups of order 8 up to gauge equivalence of braided quasi-Hopf algebras that was previously done in [6] and [5]. 相似文献
15.
ABSTRACT Let ? be a complete set of Sylow subgroups of a finite group G, that is, ? contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup H of a finite group G is said to be ?-permutable if H permutes with every member of ?. The purpose of this article is to study the influence of ?-permutability of all maximal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of a finite group G on the structure of G. Our results improve and extend the main results of Asaad (1998), Asaad and Heliel (2003), Asaad et al. (1991), Li et al. (2003), Ramadan (1992), and Srinivasan (1980). 相似文献
16.
V. V. Bavula 《代数通讯》2013,41(4):1381-1406
ABSTRACT In Dixmier (1968), the author posed six problems for the Weyl algebra A 1 over a field K of characteristic zero. Problems 3, 6, and 5 were solved respectively by Joseph (1975) and Bavula (2005a). Problems 1, 2, and 4 are still open. In this article a short proof is given to Dixmier's problem 6 for the ring of differential operators 𝒟 (X) on a smooth irreducible algebraic curve X. It is proven that, for a given maximal commutative subalgebra C of 𝒟 (X), (almost) all noncentral elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent; (ii) weakly nilpotent; (iii) generic; (iv) generic, except for a subset K*a + K of strongly semi-simple elements; (iv) generic, except for a subset K*a + K of weakly semi-simple elements, where K* := K\{0}. The same results are true for other popular algebras. 相似文献
17.
In 1967, Shioda [20] determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in [20] are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field k, char k ≠ 2, 3, 5, 7. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants. 相似文献
18.
19.
It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975), such a ring is a direct sum of an Artinian ring and hereditary prime rings. The aim of this note is to show that every serial ring having the restricted minimum condition is Noetherian. 相似文献
20.
A. R. Chekhlov 《代数通讯》2013,41(12):5059-5073
We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFI-group is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13] and [14]. 相似文献