We give a description of the Linnell division ring of a countable residually (poly- Z \mathbb {Z} virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group G G with coefficients in this Novikov ring implies the existence of a normal subgroup N ⩽ G N \leqslant G such that c d Q ( N ) > c d Q ( G ) cd_\mathbb {Q}(N) > cd_\mathbb {Q}(G) and G / N G/N is poly- Z \mathbb {Z} virtually nilpotent. As a consequence, we show that if G G is an RPVN group of finite type, then its top-degree ℓ 2 \ell ^2 -Betti number vanishes if and only if there is a poly- Z \mathbb {Z} virtually nilpotent quotient G / N G/N such that c d Q ( N ) > c d Q ( G ) cd_\mathbb {Q}(N) > cd_\mathbb {Q}(G) . In particular, finitely generated RPVN groups of cohomological dimension 2 2 are virtually free-by-nilpotent if and only if their second ℓ 2 \ell ^2 -Betti number vanishes, and therefore 2 2 -dimensional RPVN groups with vanishing second ℓ 2 \ell ^2 -Betti number are coherent. As another application, we show that if G G is a finitely generated parafree group with c d ( G ) = 2 cd(G) = 2 , then G G satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.

On Hermitian factorizations of matrices over division rings

Images of polynomial maps and the Ax-Grothendieck theorem over algebraically closed division rings

The central Nullstellensatz over centrally algebraically closed division rings

A near-field is an algebraic structure akin to a division ring where distributivity is relaxed on one side. The smallest such object is the Dickson near-field of order nine. We lay the foundations of linear codes over that near-field. This includes a systematic form for their generator matrices based on a one-sided Gauss pivot algorithm. Dual codes are defined and a formula is given for the parity check matrix. LCD codes are characterized by a direct sum property. Self-orthogonal codes are classified in short lengths.

Resultants of skew polynomials over division rings

The complete version of Kursov's theorem for matrices over division rings

Identities with inverses on matrix rings over a division ring of characteristic two

Let [Formula: see text] be a division ring with center [Formula: see text] and multiplicative group [Formula: see text], of char [Formula: see text], and with an involution *. Let [Formula: see text] be the group of unitary units of [Formula: see text], namely [Formula: see text]. We investigate various instances where the dimension [Formula: see text], and in which every non-central normal subgroup [Formula: see text] contains a free non cyclic subgroup. Among them, we consider the cases where [Formula: see text] is either generated over its center of characteristic 0 by a torsion free nilpotent group, or [Formula: see text] is the field of fractions of a group algebra [Formula: see text] of the residually torsion free nilpotent group [Formula: see text] over the field [Formula: see text] of characteristic 0 , or the field of fractions of the enveloping algebra of a locally solvable residually nilpotent Lie [Formula: see text]-algebra [Formula: see text].

We provide counter examples of finite near ring, which is not a ring. It extends the result to finite division rings. We provide counter example of finite division ring, which is not a field. It contradicts the Wedderburn's Little Theorem, asserts that all finite division rings are commutative and therefore finite field

Let k be a field of any characteristic and let Λ be a finite dimensional k -algebra. We prove that if V is a finite dimensional right Λ -module that lies in the mouth of a stable homogeneous tube T of the Auslander-Reiten quiver Λ with End ¯ Λ ( V ) a division ring, then V has a versal deformation ring R ( Λ , V ) isomorphic to k [ [ t ] ] . As a consequence we obtain that if k is algebraically closed, Λ is a symmetric special biserial k -algebra and V is a band Λ -module with End ¯ Λ ( V ) ≅ k that lies in the mouth of its homogeneous tube, then R ( Λ , V ) is universal and isomorphic to k [ [ t ] ] .

This article studies the factorization of Hermitian matrices over a division ring based on their trace values. The main result shows that the trace determines the minimum number of Hermitian factors required in factorization depending on the trace value, namely four factors are sufficient for matrices with zero trace, while for matrices with nonzero trace, one additional diagonalizable factor is required. This result indicates a relationship between additive invariance (trace) and multiplicative invariance (Dieudonné determinant) in noncommutative linear algebra.

Class A penicillin-binding proteins (aPBPs) are involved in the biosynthesis and remodelling of peptidoglycan (PG). The human bacterial pathogen Streptococcus pneumoniae produces three aPBPs, which are regulated to maintain the bacterium's ovoid shape. Evidence suggests that PBP1a and PBP2a activities are closely coordinated; however, their precise functions remain unclear. Here we characterized the pneumococcal S protein, which contains a LysM-PG-binding domain and a GpsB-interacting domain. Using S protein fusion constructs or mutant bacterial strains, we show that S protein localizes to the division ring and is required to prevent premature cell lysis and minicell formation due to aberrant division site placement. S protein interacts with PBP1a and activates its PG synthesis activity. Co-immunoprecipitation experiments combined with biochemical, genetic, structural prediction and microscopy analyses suggest that S protein is part of a larger multiprotein complex containing aPBPs and PG-modifying enzymes, and coordinated by the scaffolding protein GpsB. Together, these findings suggest that a GpsB-associated complex orchestrates PG biosynthesis and remodelling in S. pneumoniae.

In this study, we introduce NP u -digital groups based on the real quaternions by formulating them as pointed digital images in the set Z 4 of all lattice points in the four-dimensional Euclidean space R 4 . To facilitate the algebraic structure, we define a notion of digital multiplications on these pointed digital images, inspired by the multiplicative operation in the division ring of real quaternions, thereby enabling the construction of NP u -digital groups for u ∈ { 1 , 2 } . More specifically, we construct a group X of order 48 as a pointed digital image which is not an NP u -digital group, and establish an isomorphism between the group X as a pointed digital image and the binary octahedral group G, which is a well-known non-abelian group of order 48 in classical algebra. We enumerate 35 specific subgroups of X, and then investigate these subgroups thoroughly to be qualified (or non-qualified) as NP u -digital groups by explicitly listing and analyzing each of the thirty five representatives with respect to various adjacency relations κ v on Z 4 for u ∈ { 1 , 2 } and v ∈ { 1 , 2 , 3 , 4 } .

In this paper, we characterize regular nilpotent 2 x 2 matrices over Bezout domains and prove that they are unit-regular. We also demonstrate that nilpotent n x n matrices over division rings are unit-regular.

Invertibility is important in ring theory because it enables division and facilitates solving equations. Moreover, (nonassociative) rings can be endowed with an extra ''structure'' such as order and topology allowing more richness in the theory. The two main theorems of this article are contributions to invertibility in the context of partially ordered nonassociative rings \textit{and} Hausdorff sequentially Cauchy-complete weak-quasi-topological nonassociative rings. Specifically, the first theorem asserts that the interval $]0,1]$ in any suitable partially ordered nonassociative ring consists entirely of invertible elements. The second theorem asserts that if $f$ is a suitably generalized concept of seminorm from a nonassociative ring to a partially ordered nonassociative ring endowed with Frink's interval topology, then under certain conditions, the subset of elements such that $f(1-a) < 1$ consists entirely of invertible elements. Part of the assumption of the second theorem is that of Hausdorff sequential Cauchy-completeness of the first ring under the topology induced by the seminorm $f$ (which takes values in a partially ordered nonassociative ring endowed with Frink's interval topology). Frink's interval topology is an example of a coarse locally-convex $T_1$ topology. Moreover, to our knowledge, the topology induced by a seminorm into a partially ordered nonassociative ring has never been introduced. Some additional facts, such as the fact that the topology on a nonassociative ring $R_1$ induced by a norm into a totally ordered associative division ring $R_2$ endowed with Frink's interval topology (or equivalently, with the order topology, since the order of $R_2$ is total) is a Hausdorff locally convex quasi-topological group with an additional separate continuity property of the product, are dealt with in the second section ''Preliminaries''.

Let $D$ be a division ring with infinite center $F$; $σ$ be an anti-automorphism of $D$ and $m$ be a positive integer such that $σ^m\neq \mathrm{Id}$. In this paper, we show that if $D$ satisfies a $σ^m$-GRI, then $D$ is centrally finite.

The Multiplicative Matrix Waring Problem for Algebraically Closed Division Rings

We study whether a unital associative algebra A over a field admits a decomposition of the form A = Z ( A ) + [ A , A ] where Z ( A ) is the center of A and [A, A] denotes the additive subgroup of A generated by all additive commutators of A. Among our main considerations are the cases in which A is the matrix ring over a division ring, a generalized quaternion algebra, or a semisimple finite-dimensional algebra. We also discuss some applications that do not necessarily require the decomposition, such as the case where A is the twisted group algebra of a locally finite group over a field of characteristic zero: if all additive commutators of A are central, then A must be commutative.

As a generalization of the ordinary metric spaces, complex-valued metric spaces have significantly influenced research in fixed point theory. The idea intends to develop rational expressions that are insignificant within the context of cone metric spaces since the latter banks on the underlying Banach space which is not a division ring. This paper demonstrates a common fixed point theorem applicable to a pair of mappings that fulfill a rational type contractive condition in the setting of complex-valued metric spaces. The mappings examined in this study are expected to comply with particular metric inequalities, which broadens and includes various results from other scholars that were established for mappings in complex-valued metric spaces.