In 2014, the first two authors proved an extension to modules of a theorem of Camillo and Yu that an exchange ring has stable range 1 if and only if every regular element is unit-regular. Here, we give a Morita context version of a stronger theorem. The definition of regular elements in a module goes back to Zelmanowitz in 1972, but the notion of a unit-regular element in a module is new. In this paper, we study unit-regular elements and give several characterizations of them in terms of “stable” elements and “lifting” elements. Along the way, we give natural extensions to the module case of many results about unit-regular rings. The paper concludes with a discussion of when the endomorphism ring of a unit-regular module is a unit-regular ring.

In this paper, we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applications of importance, e.g., we obtain a characterization for the boundedness of a subsequence of partial sums with respect to Vilenkin–Fourier series of H1 martingales in terms of n's variation. The conditions given in this paper are in a sense necessary and sufficient.

In this paper, we study spectral properties and local spectral properties of ∞-complex symmetric operators T. In particular, we prove that if T is an ∞-complex symmetric operator, then T has the decomposition property (δ) if and only if T is decomposable. Moreover, we show that if T and S are ∞-complex symmetric operators, then so is T ⊗ S.

Fix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GLn(q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n2(1 – a−1) − εr and leading coefficient mr, where εr and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants.

Asymptotic triangulations can be viewed as limits of triangulations under the action of the mapping class group. In the case of the annulus, such triangulations have been introduced in K. Baur and G. Dupont (Compactifying exchange graphs: Annuli and tubes, Ann. Comb.3(18) (2014), 797–839). We construct an alternative method of obtaining these asymptotic triangulations using Coxeter transformations. This provides us with an algebraic and combinatorial framework for studying these limits via the associated quivers.

In this paper, we provide an explicit upper bound for the Loxton–Kedlaya rank of the maximal abelian extension of ℚ.

A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g., a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundles over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally finite modules over the geometric extension algebras are related by a recollement. As examples, we investigate parabolic affine nil Hecke algebras, geometric extension algebras associated with parabolic Springer maps and an example of Reineke of a parabolic quiver-graded Hecke algebra.

Let λ be a symmetric, normal sequence space equipped with a k-symmetric, monotone norm ‖.‖λ. Also, assume that (λ, ‖.‖λ) is AK-BK. Corresponding to this sequence space λ, we study compactness of the operator ideal Kλ. We proved compactness, completeness and injectivity of the dual operator ideal Kλd. We also investigate the factorization of these operators.

The study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

This paper is about the connection between certain Banach-algebraic properties of a commutative Banach algebra E with unit and the associated commutative Banach algebra C(X, E) of all continuous functions from a compact Hausdorff space X into E. The properties concern Ditkin's condition and bounded relative units. We show that these properties are shared by E and C(X, E). We also consider the relationship between these properties in the algebras E, B and $\~{B}$ that appear in the so-called admissible quadruples (X, E, B, $\~{B}$).

A Lie group is called p-regular if it has the p-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into p-regular SO(2n(p) is determined, which completes the list of (non)triviality of such Samelson products in p-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion SO(2n − 1) → SO(2n) in the sense of James at any prime p.

We consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

We study the homological algebra of bimodules over involutive associative algebras. We show that Braun's definition of involutive Hochschild cohomology in terms of the complex of involution-preserving derivations is indeed computing a derived functor: the ℤ/2-invariants intersected with the centre. We then introduce the corresponding involutive Hochschild homology theory and describe it as the derived functor of the pushout of ℤ/2-coinvariants and abelianization.

Milnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.

Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.

In this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

We determine the lower rank of the direct product of finitely many hereditarily just infinite profinite groups of finite lower rank.

In this paper, we mainly provide a categorical view on the braided structures appearing in the Hom-quantum groups. Let $\mathcal{C}$ be a monoidal category on which F is a bimonad, G is a bicomonad, and ϕ is a distributive law, we discuss the necessary and sufficient conditions for $\mathcal{C}^G_F(\varphi)$, the category of mixed bimodules to be monoidal and braided. As applications, we discuss the Hom-type (co)quasitriangular structures, the Hom–Yetter–Drinfeld modules, and the Hom–Long dimodules.

This paper deals with the Green ring $\mathcal{G}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$ with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring $\mathcal{G}(\mathcal{C})$, or the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero in K. For the Green ring $\mathcal{G}(\mathcal{C})$ itself, $\mathcal{G}(\mathcal{C})$ is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero. The second part of this paper focuses on the case where $\mathcal{C}=\text{Rep}(\mathbb {k}G)$ for a cyclic group G of order p over a field $\mathbb {k}$ of characteristic p. In this case, the Casimir number of $\mathcal{C}$ is computable and is shown to be 2p2. This leads to a complete description of the Jacobson radical of the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over any field K.