arXiv subjects

Sajani Lavanya Madasi

Publications and source records attributed to Sajani Lavanya Madasi.

4 recordsLinked to original sources

arXiv2026

Universal Operator Envelopes and Independent Arithmetic Defects of Additive Ternary Gamma-Rings

For an additive ternary $Γ$-ring $T$, we construct the associative rng generated by the two-element, two-index operators on ternary modules and compare it with its realization on the regular module. For a finite-free two-step system $T_μ=E\oplus W$, two integral matrices control complementary parts of this comparison: the middle-relation matrix $B_μ$ presents the hidden quotient occurring in the integral regular kernel, while the visible flattening $Θ_μ$ controls the visible image, its saturation and its nonflat base-change correction. The torsion cokernels of these matrices define an ordered arithmetic defect pair; this pair records selected arithmetic data and is not asserted to be the whole regular kernel. The principal result is a constructive realization theorem: for every ordered pair $(G,H)$ of finite abelian groups, there is a torsion-free two-step ternary $\mathbb Z$-ring whose visible-saturation torsion is $G$ and whose middle-relation torsion is $H$. Thus the two torsion mechanisms, including their prime supports and invariant factors, can be prescribed independently. The proof builds two rank-$(2,1)$ atomic tensor families with complementary Smith profiles and establishes an orthogonal-sum theorem for their presentation matrices. We also identify the standard base-change obstruction attached to these two matrices, describe their determinantal loci, realize classical rank-one matrix-factorization periodicity in an example not induced by an associative $\mathbb Z$-bilinear binary product on its underlying group, and classify all $256$ integral $2\times2\times2$ sign tensors by the resulting defect data. All finite calculations use exact integer arithmetic and are independently reproducible.

math.RA↗

arXiv2026

Finite diagonalizable torsors and Kummer cohomology over semiring schemes

Classical descent treats a finite diagonalizable torsor as a higher-rank locally free object. Over semirings that route is incomplete: fpqc-local freeness is not known to imply Zariski-local freeness in arbitrary rank. We show that diagonalizable symmetry bypasses this obstruction. The coordinate semialgebra splits into character pieces, each piece descends in rank one, and the torsor identity makes their multiplication maps invertible. This gives a natural equivalence between fpqc $D_X(Q)$-torsors and Picard-strong $Q$-gradings, and represents every torsor by a finite locally free morphism of rank $|Q|$. The same viewpoint proves that finite-index monomial maps of split tori are finite locally free torsors and yields matrix Kummer classification sequences. A concrete example is the nontrivial Boolean torsor $\Spec\mathbb B[v^{\pm1}]\to\Spec\mathbb B[u^{\pm1}]$, $u\mapsto v^m$, whose unit-power classes contribute $\mathbb Z/m\mathbb Z$ to degree-one cohomology. A two-term resolution of the character group then gives a presentation-independent universal-coefficient filtration for diagonalizable fpqc cohomology. This formalism organizes, but does not compute, $\mathbf G_{\mathrm m}$-cohomology. In degree two it identifies the Kummer boundary of a line bundle with its root gerbe. On projective space over the Boolean or real tropical semifield all $μ_m$-torsors vanish, whereas the root gerbe of $\mathcal O(1)$ has exact order $m$. Thus torsors and gerbes retain information that can disappear under ordinary ring completion.

math.RA↗

arXiv2026

Central Splitting, Division-Stable Residuals, and Opposite-Ring Transfer for Strongly C4*-Rings

The known decomposition theorem for strongly $\Cfourstar$-modules gives a semisimple summand and a summand-square-free residual summand, with two-way Hom-orthogonality when the ambient module is projective. Applied to the regular module, the two cross-corners vanish, so the idempotent defining the decomposition is central. Thus every strongly right $\Cfourstar$-ring splits as $R\cong Σ\times T$, where $Σ$ is semisimple artinian and $T_T$ is summand-square-free. The splitting need not be unique. We organize all admissible central splittings into a join-semilattice and prove a comparison theorem: any two residual factors have a common direct factor, and their complementary factors are finite products of division rings. Consequently the right-to-left defect is independent of the chosen splitting. If the central idempotents satisfy the ascending chain condition, there is a unique greatest admissible idempotent; it captures every central semisimple artinian direct factor and hence yields a canonical residual with no further such factor. An infinite product of fields shows that this finiteness hypothesis cannot simply be omitted. We also prove anti-isomorphism transport for $\Cfour$, $\Cfourstar$, semi-weak-CS, and strongly $\Cfourstar$ modules. It yields transfer whenever a residual factor is a product of a semisimple summand-square-free ring and a self-opposite core. This criterion does not assume regularity, exchange, or a left-sided hypothesis. Skew Laurent rings provide noncommutative nonregular examples, while the known injective-nonsurjective skew-polynomial construction gives a sharp one-sided residual obstruction.

math.RA↗

arXiv2026

Cyclic crossed-product envelopes and arity-detecting relative $K$-theory of automorphism-derived $n$-ary algebras

Let $m=n-1$. For a unital associative algebra $A$, an automorphism $σ$, and $c\in A^\times$ satisfying $σ(c)=c$ and $σ^m=\operatorname{Ad}_c$, we study the Hosszú--Gluskin-type product $[a_0,\ldots,a_m]=a_0σ(a_1)\cdotsσ^{m-1}(a_{m-1})ca_m$. We prove that its endpoint-anchored all-slot envelope is the cyclic crossed product generated by $A\otimes A^{\mathrm{op}}$ and $u$, with $ux=σ^e(x)u$ and $u^m=c\otimes(c^{-1})^{\mathrm{op}}$. For the finite-order case $c=1$ and permutation algebras $A=k^X$, this gives an orbit--stabilizer block decomposition. A general dimension-defect theorem then computes the homotopy fibre of the forgetful $K$-theory map. In characteristic zero, transitive $C_m$-sets of every divisor length $d\mid m$ have the same ordinary spectrum $\K(k)^m$ but boundary $\mathbb Z/d$. The regular action gives $π_{-1}\cong\mathbb Z/m$, recovering the arity; over finite fields, the remaining torsion is governed by multiplicative orders.

math.RA↗