Glasgow Math Research Articles

Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows

We show an analog of a theorem of [J. An, A. Ghosh, L. Guan and T. Ly, Bounded orbits of diagonalizable flows on finite volume quotients of products of [Formula: see text], Adv. Math. 354 (2019) 106743] on weighted badly approximable vectors for totally imaginary number fields. We show that for [Formula: see text] and [Formula: see text] a lattice subgroup, the points of [Formula: see text] with bounded orbits under a one-parameter Ad-semisimple subgroup of [Formula: see text] form a hyperplane-absolute-winning set. As an application, we also provide a generalization of a result of [R. Esdahl-Schou and S. Kristensen, On badly approximable complex numbers, Glasgow Math. J. 52 (2010) 349–355] about the set of badly approximable complex numbers.

A noncommutative Gretsky–Ostroy theorem and its applications

Let H \mathcal {H} be a separable Hilbert space and let B ( H ) B(\mathcal {H}) be the ∗ * -algebra of all bounded linear operators on H \mathcal {H} . In the present paper, we prove that a positive/regular operator from L 1 ( 0 , 1 ) L_1(0,1) into an arbitrary separable operator ideal in B ( H ) B(\mathcal {H}) is necessarily Dunford–Pettis, extending and strengthening results due to Gretsky and Ostroy [Glasgow Math. J. 28 (1986), pp. 113–114], and Holub [Proc. Amer. Math. Soc. 104 (1988), pp. 89–95]. Consequently, for an arbitrary atomless von Neumann algebra M \mathcal {M} and an arbitrary KB-ideal C E C_E in B ( H ) B(\mathcal {H}) , the predual M ∗ \mathcal {M}_* of M \mathcal {M} is not isomorphic to any subspace of C E C_E . This observation complements several earlier results.

A generalized Frattini subgroup

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If [Formula: see text] is a subnormal subgroup of a finite group [Formula: see text] containing [Formula: see text], then whether the supersolvability of [Formula: see text] follows the supersolvability of [Formula: see text]”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal [Formula: see text]-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On [Formula: see text]-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]). In this paper, we give the answer to the more general question: “Let [Formula: see text] be a local formation. If [Formula: see text] is a subnormal subgroup of a group [Formula: see text], then in what case [Formula: see text] will follow from [Formula: see text]”.

Ricci curvature on warped product submanifolds of complex space forms and its applications

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

R-Fredholm theory in ordered Banach algebras

The introduction of Fredholm theory relative to general unital homomorphisms $$T:A \rightarrow B$$ between Banach algebras A and B, which involves the study of Fredholm, Weyl and Browder elements, was due to Harte (Math Z 179:431–436, 1982), after which this investigation was continued by several authors. Motivated by results of Alekhno (Positivity 11:375–386, 2007, Positivity 13:3–20, 2009), the definitions of upper Weyl and upper Browder elements in an ordered Banach algebra (OBA) were given in Benjamin and Mouton (Quaest Math 39:643–664, 2016), thereby initiating the study of the interplay between Fredholm theory and ordering. Inspired by an indication that these elements could be useful in the context of OBAs, certain variants of the invertible, Fredholm, Weyl and Browder elements (namely the r-invertible, r-Fredholm, r-Weyl and r-Browder elements) were investigated in Benjamin et al. (Glasgow Math J, 2018. https://doi.org/10.1017/S0017089518000393) in the context of general Banach algebras. Continuing the development of Fredholm theory in ordered Banach algebras, we now introduce the notions of upper r-Weyl, contractive upper r-Weyl, upper r-Browder and contractive upper r-Browder elements in an OBA and, motivated by previous results, investigate conditions under which the positive almost r-invertible r-Fredholm elements will be contractive upper r-Browder. This also leads to the recovering (and strengthening) of certain results in Benjamin and Mouton (Positivity 21:575–592, 2017) regarding the upper Browder spectrum property.

GK dimension of some connected Hopf algebras

While it was identified that the growth of any connected Hopf algebras is either a positive integer or infinite, we have yet to determine the Gelfand–Kirillov (GK) dimension of a given connected Hopf algebra. We use the notion of anti-cocommutative elements introduced in Wang, D. G., Zhang, J. J., Zhuang, G. (2013). Coassociative lie algebras. Glasgow Math. J. 55(A):195–215 to analyze the structure of connected Hopf algebras generated by anti-cocommutative elements and compute the GK dimension of said algebras. Additionally, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements.

DISTRIBUTION OF GALOIS GROUPS OF MAXIMAL UNRAMIFIED 2-EXTENSIONS OVER IMAGINARY QUADRATIC FIELDS

Let$k$be an imaginary quadratic field with$\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert$2$-class field tower is at least$2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J.40(1) (1998), 63–69) calculated the density of$k$where the length of the tower is$1$; that is, the maximal unramified$2$-extension is a$V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

FP $$_n$$ n -Injective and FP $$_n$$ n -Flat Complexes

Let R be an arbitrary ring and $$n\ge 0$$ be an integer or $$n = \infty $$ . We introduce and study FP $$_n$$ -injective and FP $$_n$$ -flat complexes of modules, which unify the following notions: (FP-)injective and flat complexes (Yang and Liu in Commun Algebra 38:131–142, 2010; Enochs and Garcia Rozas in J Algebra 210:86–102, 1998), absolutely clean and level complexes (Bravo and Gillespie in Commun Algebra 44:2213–2233, 2016) and weak injective and weak flat complexes (Gao and Huang in Glasgow Math J 58:539–557, 2016). Suppose that $$n>1$$ is an integer. It is shown that a complex C is FP $$_n$$ -injective (resp. FP $$_n$$ -flat) if and only if C is exact and all cycles of C are FP $$_n$$ -injective (resp. FP $$_n$$ -flat) as R-modules. Then we investigate duality pairs relative to the FP $$_n$$ -injective and FP $$_n$$ -flat complexes. It is shown that the pairs (fp $$_n\mathcal {I}$$ , fp $$_n\mathcal {F}$$ ) and (fp $$_n\mathcal {F}$$ , fp $$_n\mathcal {I}$$ )) are duality pairs, where fp $$_n\mathcal {I}$$ and fp $$_n\mathcal {F}$$ denote the subcategories of FP $$_n$$ -injective and FP $$_n$$ -flat complexes, respectively. As applications, we get that any complex admits an FP $$_n$$ -injective (resp. FP $$_n$$ -flat) cover and preenvelope. In addition, cotorsion pairs associated with the FP $$_n$$ -injective and FP $$_n$$ -flat complexes are considered.

On a theorem of Brian Fisher in the framework of w-distance

In 1980. Fisher in [Fisher, B., Results on common fixed points on complete metric spaces, Glasgow Math. J., 21 (1980), 165–167] proved very interesting fixed point result for the pair of maps. In 1996. Kada, Suzuki and Takahashi introduced and studied the concept of w–distance in fixed point theory. In this paper, we generalize Fisher’s result for pair of mappings on metric space to complete metric space with w–distance. The obtained results do not require the continuity of maps, but more relaxing condition (C; k). As a corollary we obtain a result of Chatterjea.

HOMOLOGICAL DIMENSIONS OF CROSSED PRODUCTS

AbstractIn this paper, we consider several homological dimensions of crossed products AασG, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of AσαG are classified: global dimension of AσαG is either infinity or equal to that of A, and finitistic dimension of AσαG coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S ≤ G, we show that A and AασG share the same homological dimensions under extra assumptions, extending the main results in (Li, Representations of modular skew group algebras, Trans. Amer. Math. Soc.367(9) (2015), 6293–6314, Li, Finitistic dimensions and picewise hereditary property of skew group algebras, to Glasgow Math. J.57(3) (2015), 509–517).

ON THE p-ADIC BINOMIAL SERIES AND A FORMAL ANALOGUE OF HILBERT'S THEOREM 90 – ADDENDUM

AbstractThe proof of Theorem 3.2 in [1] (P. Tzermias, On the p-adic binomial series and a formal analogue of Hilbert's Theorem 90, Glasgow Math. J.47 (2005), 319–326) contains two opaque claims. The necessary clarifications are provided here.

Q-DEDEKIND-TYPE DAEHEE-CHANGHEE SUMS WITH WEIGHT α ASSOCIATED WITH MODIFIED q-EULER POLYNOMIALS WITH WEIGHT α

Recently, q-Dedekind-type sums related to q-Euler polynomials was studied by Kim in [T. Kim, Note on q-Dedekind-type sums related to q-Euler polynomials, Glasgow Math. J. 54 (2012), 121-125]. It is aim of this paper to consider a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight related to modified q-Euler polynomials with weight by using Kim's p-adic q-integral.

SUBMODULES OF COMMUTATIVE C*-ALGEBRAS

AbstractIn this paper we generalise a result of Izuchi and Suárez (K. Izuchi and D. Suárez, Norm-closed invariant subspaces in L∞ and H∞, Glasgow Math. J. 46 (2004), 399–404) on the shift invariant subspaces of $L^\infty(\mathbb{T})$ to the non-commutative setting. Considering these subspaces as $C(\mathbb{T})$-modules contained in $L^\infty(\mathbb{T})$, we show that under some restrictions, a similar description can be given for the ${\mathfrak{B}}$-submodules of ${\mathfrak{A}}$, where ${\mathfrak{A}}$ is a C*-algebra and ${\mathfrak{B}}$ is a commutative C*-subalgebra of ${\mathfrak{A}}$. We use this to give a description of the $\mathbb{M}_n({\mathfrak{B}})$-submodules of $\mathbb{M}_n({\mathfrak{A}})$.

A class of unary semigroups admitting a Rees matrix representation

In Billhardt et al. (Semigroup Forum 79:101–118, 2009) the authors introduced the notion of an associate inverse subsemigroup of a regular semigroup, extending the concept of an associate subgroup of a regular semigroup, first presented in Blyth et al. (Glasgow Math. J. 36:163–171, 1994).

A note on the Diophantine equation $$y^2=px(A{x^{2}}+2)$$

Recently, Li Min Chen (Acta Math Sinica Chin Ser 53:83–86, 2010) considered a variant \(y^2=px(x^2+2)\) of Cassels’ equation \(y^2=3x(x^2+2)\). He proved that the equation has at most two solutions in positive integers \((x, y)\). Therefore, he improved a result obtained Luca and Walsh (Glasgow Math J 47:303–307, 2005). In this note, we consider another variant of Cassels’ equation and we show that for any prime \(p\) and any odd positive integer \(A\), the Diophantine equation \(y^2=px(Ax^2+2)\) has at most seven solutions in positive integers \((x, y)\).

ON WEAK ZIP SKEW POLYNOMIAL RINGS

We introduce the notion of nil(α, δ)-compatible rings which is a generalization of reduced rings and (α, δ)-compatible rings. In [Ore extensions of weak zip rings, Glasgow Math. J.51 (2009) 525–537] Ouyang introduces the notion of right (respectively, left) weak zip rings and proved that, a ring R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip, when R is (α, δ)-compatible and reversible. We extend this result to the more general situation that, when R has (α, δ)-condition and quasi-IFP, then nil (R)[x; α, δ] = nil (R[x; α, δ]); and R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip.

Simplicial cohomology of band semigroup algebras

We establish the simplicial triviality of the convolution algebra l1 (S), where S is a band semigroup. This generalizes some results of Choi (Glasgow Math. J. 48 (2006), 231–245; Houston J. Math. 36 (2010), 237–260). To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on l(S). Crucial to our approach is the use of the structure semilattice of S, and the associated grading of S, together with an inductive normalization procedure in cyclic cohomology. The latter technique appears to be new, and its underlying strategy may be applicable to other convolution algebras of interest.

A note on two equivalence relations on inverse semigroups

Let S be an inverse semigroup. In Rezavand et al. (Semigroup Forum 77:300–305, 2008) and Munn (Proc. Glasgow Math. Assoc. 5:41–48, 1961) two equivalence relations are defined on S. After describing these relations we show that they coincide.

MONOLITHIC MODULES OVER NOETHERIAN RINGS

AbstractWe study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.

ON RADICAL FORMULA IN MODULES

AbstractWe will state some conditions under which if a quotient of a module M satisfies the radical formula of degree k (s.t.r.f of degree k), so does M. Especially, we will introduce some particular modules M′ such that M′ ⊕ M″ s.t.r.f of degree k, when M″ s.t.r.f of degree k. Furthermore, we will show that, under certain conditions, if the completion of a module M s.t.r.f of degree k, then there is a non-negative integer k′ such that M s.t.r.f. of degree k′. Moreover, we state a corrected version of Leung and Man's theorem (K. H. Leung and S. H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), 285–293) on Noetherian rings that satisfies the radical formula.