arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 8 new papers, 4 cross-lists and 4 replacements in Rings and Algebras (math.RA), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We prove that any variety of associative algebras over a Noetherian commutative ring with unity satisfying the identity x^3=0 is finitely based. This property has been known for the varieties satisfying the identity x^2=0. On the other hand, there are varieties satisfying the identity x^6=0 that are not finitely based.
We study rings over which every semiprime submodule of every module is an intersection of prime submodules. This holds if semiprime two-sided ideals satisfy ACC, if principal two-sided ideals are finitely generated as left ideals, or if the ring is PI. Over Jacobson PI rings, every finitely generated module is Jacobson: its prime submodules are intersections of maximal submodules. For affine PI algebras this extends the Amitsur-Procesi theorem to modules. In every ring with involution, real two-sided ideals are intersections of real prime two-sided ideals. The same holds for real submodules of modules over PI rings and hereditary Noetherian prime rings, but can fail for real left ideals in Noetherian domains. Formally real affine PI algebras need not be real Jacobson: their real prime two-sided ideals need not be intersections of real primitive ideals. We give a sufficient positivity criterion. Over real Jacobson PI rings, finitely generated modules are real Jacobson: their real prime submodules are intersections of real maximal submodules. Over finitely generated free associative algebras, every finitely generated semiprime submodule of a finite free module is a finite intersection of finitely generated prime submodules and an intersection of maximal submodules of finite codimension. Semiprime hulls of finitely generated submodules are finitely generated. Evaluation at finite-dimensional irreducible representations and nonzero vector tuples gives a Hilbert Nullstellensatz over arbitrary fields. For free algebras with involution, the corresponding real results are known over the real or complex numbers when the involution permutes the generators. We extend real prime decomposition to arbitrary fields and involutions, and give new algebraic proofs of the real Nullstellensatz and finite generation of real radicals, without the GNS construction or Groebner bases.
We study four-dimensional binomial Artin-Schelter regular algebras and their connection with twisted monoid algebras. Starting with $23$ examples up to permutation of generators, we investigate the associated monoids of I-type and the structure of their groups of fractions. We determine the corresponding finite permutation groups and compute the second group cohomology $H^2(G(M),k^\times)$. These computations allow us to construct explicit families of four-dimensional Koszul Artin-Schelter regular algebras through cocycle twists. We illustrate this construction for some of these algebras.
We describe Hochschild cohomology of Temperley--Lieb algebras by explicit minimal-resolution coordinates and multilinear operations on Kauffman diagrams. For each doubled-line basic algebra, a finite four-term bimodule complex is verified by a signed-shuffle comparison and an augmented contracting homotopy. We compute cocycles and coboundaries and recover the known cup product and Gerstenhaber bracket in explicit representatives. Over $\mathbb C$, at nonzero root-of-unity loop parameters, we construct a tower-compatible derivation and the marked-circle two-cocycle. Their classes are nonzero on every nonsemisimple block; their cup products therefore represent all positive classes over the centre. Relative full twists give a compact expression for the derivation. Worked planar diagrams distinguish the cup product from insertion and strict cochain equalities from equalities of classes. A finite-lattice section interprets the marked-circle operation as loop-weight response, derives its planar Potts normalization, and exhibits a six-vertex realization. It also shows why passing to a deformable spin representation may annihilate the class with different coefficients. No thermodynamic conclusion is claimed, and the even-strand zero-parameter family remains separate.
Given a finitely aligned higher-rank $k$-graph $\Lambda$, Farthing, Muhly and Yeend (Semigroup Forum 2005) constructed an inverse semigroup $S_\Lambda$ to provide a groupoid model for the $C^*$-algebra of $\Lambda$. In this paper, we extend some of the existing results related to the semigroup structure of a graph inverse semigroup to the more general setting of the inverse semigroup $S_\Lambda$. We obtain geometric descriptions of the Green's relations on $S_\Lambda$, which effectively generalize the existing characterization of Green's relations on a graph inverse semigroup. We provide a simple and useful description of the Cuntz-Krieger ideal of the Booleanization of $S_\Lambda$ and, based on this, identify a certain $X$-to-join Booleanization of $S_\Lambda$, where the abstract and concrete Cuntz-Krieger relations agree. Besides the natural partial order, $S_\Lambda$ is also ordered by set-inclusion. This helps us to realize $S_\Lambda$ as a semigroup \emph{graded-ordered} by $\mathbb{Z}^k$ and, subsequently, gives rise to a full inverse subsemigroup $S_\Lambda^h$ of $S_\Lambda$, which is graded by $\mathbb{Z}^k$ in the usual sense. We show that the Kumjian-Pask algebra of $\Lambda$ is graded isomorphic to the quotient of the contracted semigroup algebra of $S_\Lambda^h$ by the tight ideal. In the final part of the paper, we show that a $k$-graph morphism $p:\Lambda_1\longrightarrow \Lambda_2$ that is injective and satisfies a certain path lifting property induces, in a natural way, an injective $0$-preserving semigroup homomorphism from $S_{\Lambda_1}$ to $S_{\Lambda_2}$. In view of this, we build a suitable categorical setup to conclude that the isomorphism established between the Kumjian-Pask algebra of $\Lambda$ and the tight ideal quotient of the semigroup algebra of $S_\Lambda^h$ is natural.
For two third-order tensors over $\mathbb{R}$ or $\mathbb{C}$, we define the Strassen CP rank gap as the sum of their CP ranks minus the CP rank of their direct sum. For arbitrary third-order tensors, we derive explicit upper bounds on this gap in terms of CP rank excess and dimension. For symmetric third-order tensors, we similarly define the Strassen symmetric rank gap using symmetric rank and establish explicit upper bounds in terms of symmetric rank excess, CP rank excess, and dimension. These bounds also yield sufficient conditions for the respective gaps to vanish, ensuring additivity of CP rank or symmetric rank under direct sums.
A Kelmans transformation increases or preserves the largest eigenvalue of a symmetric nonnegative matrix. We determine when this comparison holds for Hermitian block extensions with a fixed coupling and an arbitrary common complementary block. An orthogonal change of basis and Schur complementation give an exact formula for the compressed resolvent difference as a difference of two rank-one matrices. This formula characterizes admissible coupling subspaces by proportionality of projected resolvent vectors. Polynomial coefficient kernels describe all such subspaces, and cyclic subspaces give dimension bounds and a maximum-dimension result for a two-block family. We also prove a general block comparison theorem and give monotonicity and equality criteria for nonnegative rank-one couplings. Applications concern graph joins, complex attachments, and Hermitian adjacency matrices.
Using Mac Lane's PROPs, one can introduce multilinear polynomial identities that involve all the operations of a Hopf algebra: the (co)multiplication, the (co)unit and the antipode. We find bases for polynomial identities and calculate their codimensions for the Sweedler algebra, treating it first as an ordinary associative algebra over a field and second as a Hopf algebra.
Although non-dimensionalisation is a fundamental step in modelling, it is rarely obvious, especially for large dynamical systems, how to systematically construct optimal non-dimensionalisations. In this work, we develop a general mathematical framework for non-dimensionalising polynomial ODEs by interpreting them as chemical reaction networks (CRNs) and characterise the space of admissible scalings. Within this viewpoint, admissible scalings of time and species are encoded by Laurent monomials, and we derive exact formulae for the minimal number of multiplicative dimensionless parameters required by the reaction-channel coefficients and by the full initial value problem, together with characterisations of the scale assignments that attain structural lower bounds. These counts are expressed in terms of structural properties of the underlying CRN, and we show that the full initial value problem requires no more parameters than the corresponding Buckingham count. These results lead to constructive algorithms which we analyse rigorously and implement in the open-source Python package CRNNondim, available at https://github.com/georgio-hawi/crnnondim . Given a polynomial system or CRN, CRNNondim determines species and time scalings that bring variables as close to unity as possible while minimising the number of free parameters, and among the admissible scalings it chooses free parameters that result in the corresponding dimensionless system being sparse and simple. We apply CRNNondim to numerous well-known biological and biochemical models, yielding simpler dimensionless systems with fewer free parameters, and interpreting the parameters that govern their dynamics.
For every finite-dimensional simple Euclidean Jordan algebra $V$ of rank at least three, we realize the function $\operatorname{tr}(x^3)/|x|$ on its traceless subspace $V_0$ as a $C^{1,1}\setminus C^2$ viscosity solution of a uniformly elliptic equation $F(D^2 u)=0$ in $\mathbb{R}^{\operatorname{dim}V_0}$. This unifies and extends earlier constructions from Cartan isoparametric cubics, Hsiang minimal cubics, and traceless real symmetric matrices.
Do Kaplansky's group-ring conjectures hold for torsion-free word-hyperbolic groups? We show that all four fail in positive characteristic. For every prime $p$ there is a finitely presented, torsion-free, word-hyperbolic group $G$ with a finite two-dimensional classifying space and $a,b\in\mathbb{F}_p[G]$ with $ab=1\neq ba$. So, over every field of characteristic $p$, the group algebra of $G$ is not directly finite and has a non-trivial idempotent, zero divisor and unit, and $G$ is neither sofic nor surjunctive. For $p=2$, $G$, $a$ and $b$ come from OpenAI's random construction, and we add that $G$ is hyperbolic with probability $1-o(1)$, as are OpenAI's zero-divisor groups. For odd $p$ the groups are new: they adapt OpenAI's direct-finiteness construction with a signed cancellation, found first by S. V. Ivanov and independently by us, via an exact solution of a colour-balance problem and a Perron certificate. As far as we know, they are the first torsion-free counterexamples to direct finiteness and to the idempotent conjecture in odd characteristic. Hyperbolicity comes from one deterministic criterion: if a labelled graph is pattern-free, the group obtained by coning off its components is aspherical and has a linear isoperimetric inequality. We prove it by running OpenAI's exclusion of reduced spheres on van Kampen discs, an accounting OpenAI used for another complex. For every prime $p$, the existence of $G$, $a$ and $b$ and the failure of all four conjectures are formalized in Lean 4, including existence forms of the random-graph estimates used. In characteristic zero, where the idempotent and direct-finiteness conjectures hold for torsion-free hyperbolic groups, we show that no cancellation built into the vertex types survives the decay estimate that makes the groups torsion-free. OpenAI's characteristic-two counterexample with torsion is a CAT(0) group.
Let $R$ be a commutative noetherian ring. We ask whether complexes with finitely generated cohomology and a fixed support can be generated by relatively compact objects, using coproducts which exist in the half-bounded derived category. We prove that $\mathbf D^+_{\mathrm{fg},W}(R)=\operatorname{Loc}_{\mathrm{ex}}(\mathbf D^b_{\mathrm{fg},W}(R))$ whenever $W\subseteq\operatorname{Spec} R$ is specialization closed and has finite dimension. The same equality holds for every specialization-closed subset of $\operatorname{Spec} B$ when $B$ is a finite-type algebra over a noetherian domain $A$ of finite character and either $A$ or $B$ is regular. These equalities classify the thick Hom-submodules closed under existing coproducts. We also prove the corresponding bounded-above statement from the support results of Matsui--Takahashi. For an arbitrary noetherian ring, the bounded-below generation conjecture reduces to torsion complexes over quotient domains, with their support unchanged. Finally, over a regular noetherian domain of infinite dimension, we give a bounded-below object which is generated by bounded objects with the same support but has no bounded-below representative by finitely generated modules. The comparison functor from the bounded-below derived category of finite modules is therefore neither full nor essentially surjective.