We prove that a Malcev algebra [Formula: see text] that belongs to a certain variety of Malcev algebras [Formula: see text] such that [Formula: see text] contains the [Formula: see text]-dimensional exceptional Malcev algebra [Formula: see text] is isomorphic to a direct sum [Formula: see text], where [Formula: see text] is a commutative associative algebra. Also, we prove that a Malcev superalgebra [Formula: see text] in [Formula: see text] whose even part [Formula: see text] contains [Formula: see text] is isomorphic to a direct sum [Formula: see text], where [Formula: see text] is a supercommutative associative superalgebra.

This study introduces and investigates the idea of $S$-pm-rings, a generalization of pm-rings in the context of commutative rings with a multiplicatively closed subset $S$. We prove that a ring $R$ is an $S$-pm-ring if and only if its $S$-maximal spectrum is a retract (specifically, a deformation retract) of its $S$-prime spectrum. Furthermore, we establish the equivalence of the $S$-pm-ring property to the normality of the $S$-prime spectrum and the Hausdorff property of the $S$-maximal spectrum. We also explore the relationship between $S$-pm-rings and $S$-clean rings, demonstrating that every $S$-local ring is $S$-clean, and every $S$-clean ring is an $S$-pm-ring. These results extend classical theorems in commutative algebra and algebraic geometry to the $S$-version context.

In this paper, an infinite-dimensional holomorphic functional calculus is established for vector-valued function algebras. Let E be a commutative unital Banach algebra, A be an admissible E-valued function algebra, and . Denoted by , the E-valued spectrum of f is defined as a compact subset of E. It is shown that if a function is holomorphic on a neighbourhood then . As a result, it is proved that the only admissible E-valued uniform algebras are of the form where A runs over the complex uniform algebras.

Metal–organicframeworks (MOFs) are a class of importantcrystalline and highly porous materials whose hierarchical geometryand chemistry hinder interpretable predictions in materials properties.Commutative algebra is a branch of abstract algebra that has beenrarely applied in data and material sciences. We introduce the firstever commutative algebra modeling and prediction in materials science.Specifically, category-specific commutative algebra (CSCA) is proposedas a new framework for MOF representation and learning. It integrateselement-based categorization with multiscale algebraic invariantsto encode both local coordination motifs and global network organizationof MOFs. These algebraically consistent, chemically aware representationsenable compact, interpretable, and data efficient modeling of MOFproperties such as Henry’s constants and uptake capacitiesfor common gases. Compared to traditional geometric and graph-basedapproaches, CSCA achieves comparable or superior predictive accuracywhile substantially improving interpretability and stability acrossdata sets. By aligning commutative algebra with the chemical hierarchy,the CSCA establishes a rigorous and generalizable paradigm for understandingstructure and property relationships in porous materials and providesa nonlinear algebra-based framework for data-driven material discovery.

This article provides a rigorous exploration of the transition from classical Cartesian coordinate systems to abstract geometric frameworks. It begins by establishing the “death of the fixed origin” arguing that modern analytical geometry is better understood through the lens of Commutative Algebra and Topology rather than simple numerical plotting. The text covers three major theoretical shifts: the development of Algebraic Varieties and Coordinate Rings, the introduction of Scheme Theory by Alexander Grothendieck, and the application of Sheaf Theory to maintain global consistency in complex manifolds. By synthesizing these high-level concepts, the article demonstrates how abstract geometry serves as the underlying language for both theoretical physics (specifically String Theory) and modern data science. As well as the article is designed for an advanced undergraduate or graduate-level audience. It successfully bridges the gap between pedagogical geometry and contemporary research. A particular strength of the piece is its treatment of Hilbert’s Nullstellensatz, which it uses to prove the fundamental link between algebraic ideals and geometric shapes. The inclusion of Differential Geometry and the Metric Tensor provides a holistic view, ensuring the reader understands both the algebraic and the continuous aspects of the field.

Abstract We construct C*-dynamical systems for the dynamics of classical infinite particle systems describing harmonic oscillators interacting with arbitrarily many neighbors on lattices, as well on more general structures. Our approach allows particles with varying masses, varying frequencies, irregularly placed lattice sites and varying interactions subject to a simple summability constraint. A key role is played by the commutative resolvent algebra, which is a C*-algebra of bounded continuous functions on an infinite-dimensional vector space, and in a strong sense the classical limit of the Buchholz–Grundling resolvent algebra, which suggests that quantum analogs of our results are likely to exist. For a general class of Hamiltonians, we show that the commutative resolvent algebra is time-stable and admits a time-stable sub-algebra on which the dynamics is strongly continuous, therefore obtaining a C*-dynamical system.

Abstract Tambara functors are an equivariant generalization of commutative rings. In previous work, Nakaoka introduced the spectrum of prime ideals of a Tambara functor and computed the spectrum of the Burnside Tambara functor, the equivariant analogue of the Zariski spectrum of the integers, over cyclic $p$-groups. Subsequently, Calle and Ginnett computed the spectrum of the Burnside Tambara functor over arbitrary finite cyclic groups using a generalization of Dress’ ghost coordinates for Burnside rings. In this paper, we compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group. Our proof uses recent advances in the commutative algebra of Tambara functors, as well as a Tambara functor analogue of ghost coordinates, which works over arbitrary finite groups and clarifies some previous computations. As examples, we explicitly compute the spectrum of the Burnside Tambara functor over all dihederal groups, the quaternion group $Q_{8}$, the alternating group $A_{4}$, and the general linear group $GL_{3}(\mathbb{F}_{2})$.

In this paper, we introduce the concept of a weighted matching tridendriform algebra, which generalizes both the tridendriform algebras introduced by Loday and Ronco and the [Formula: see text]-tridendriform algebras proposed by Burgunder and Ronco. Furthermore, we construct a free commutative matching tridendriform algebra using the matching augmented shuffle product. Finally, we introduce the concept of universal enveloping commutative matching Rota–Baxter algebras of commutative matching (tri)dendriform algebras and give a construction of the universal enveloping algebras.

In the forty years of existence of derived category, it was first thought as a tool in algebraic geometry, especially in the development of duality theories that were done by Hartshorne and others. After these first moments, the theory provided a powerful homological tool for the study of linear differential equations. The basic example in the literature that can be found about this is the Riemann-Hilbert problem of associating suitable regular systems of differential equations to constructible sheaves. This studies can be found in the work of Kashiwara and Schapira. See M. Kashiwara, P. Schapira “Sheaves on manifolds" ([15]). To understand the structure of the derived category is necessary to study the axioms of triangulated categories that were introduced in the mid 1960’s by J.L.Verdier in his thesis “Des catégories dérivées des catégories abéliennes" ([21]). The role of the triangles in the derived category is a similar role of the exact sequence in the abelian category. But it is important to remember that these axioms had their origins in algebraic geometry and algebraic topology. Nowadays there are important applications of triangulated categories in areas like algebraic geometry, algebraic topology (stable homotopy theory), commutative algebra, differential geometry and representation theory of artin algebras. See, for instance, the book of D. Happel- “Triangulated categories and the representation theory of finite dimensional algebras" ([11]). The objective of this notes is to present an introdutory material to the undergraduate and graduate students that would like to know some ideas about the derived category. These are the notes a one week series of introductory lectures which I gave in the XXIII-Escola de Algebra, in Maringá, Paraná, Brazil. Firstly we introduced the concepts of additive and abelian category to show the axioms of triangulated category that are our main objective. The triangulated category obey four axioms. We first introduced the first three axioms and their consequences on chapter one and then the octahedral axioms in various equivalent forms in a separate section of the first chapter. The objective of this section is to give a model capable of making this axiom more palatable since, in general, the form that it is presented in the literature does not remind the reader of any similar structure in other fields of mathematics. So, we make the necessary efforts here to present another form of this axiom that is similar to other tools that could be seen in the abelian categories. We present in chapter one the main example of triangulated category, the homotopy category of complexes. Secondly, to understand the morphisms in the derived category I introduced the concept of localization in chapter two. To those that are starting to study localization, we present the necessary background to understand the localization of non commutative ring. We believe that with this model in mind the student will profit more from the study of localization of categories. On chapter two, the student will find the necessary information and exercises to begin to manipulate morphisms in the derived category. So, on chapter three we introduce the definition of derived category of an abelian category and we explain how one sees the original abelian category as a subcategory of its derived category. After having done all this work, it is natural to have many questions about the behavior of the derived category or its applications. Therefore, we present here a bibliography in portuguese and in english that will help the students to make further investigations. The reader that whishes to know the history and the motivation of the begining of the derived category with many details, should read the introduction of the book "Sheaves on Manifolds - M. Kashiwara and M. Schapira ([15]). Acknowledgements: I am particularly grateful to Sônia Maria Fernandes-DMAUFV, Tanise Carnieri Pierin -DMAT-UFPR and Eduardo Nascimento Marcos IMEUSP, who carefully worked through the text and sent me detailed lists of corrections, questions and remarks. These notes were writen for the first time in 2014 and were used in a minicourse which I tough in the XXIII-Escola de Algebra in Maringá, Paraná, Brazil. The last version was written during my visit to IME-USP in 2018, where I got finantial help of Fapesp, process 2018/08104 - 3.

This article concerns the notion of weak circular minimality being a variant of o-minimality for circularly ordered structures. We consider the binary level of these structures forming algebras of binary isolating formulas, which are based on families of labels and compositions of related formulas. These algebras are studied for ℵ0-categorical 1-transitive non-primitive weakly circularly minimal theories of convexity rank greater than 1 with a trivial definable closure having a non-trivial monotonic-to-right function to the definable completion of a structure. On the basis of the study, the authors present a description of these algebras. It is shown that for this case there exist only commutative algebras. A strict 𝑠-deterministicity of such algebras for some natural number 𝑠 is also established.

Quasiorthogonality of commutative algebras, complex Hadamard matrices, and mutually unbiased measurements

The cone of a classical group \(G\) is an affine \(G\times G\)-variety. The aim of this note is to initiate its combinatorial study in the cases when \(G\) is the complex orthogonal or symplectic group (the case of the general linear group being well documented in the literature). The coordinate ring of the cone of G is a finitely generated commutative graded algebra. First the \(G\times G\)-module structure of its homogeneous components is determined. This is used to compute the Hilbert series of this coordinate ring in the cases when G is the orthogonal group \(\mathrm{O}(3)\), \(\mathrm{O}(4)\), the special orthogonal group \(\mathrm{SO}(4)\), and when \(G\) is the symplectic group \(\mathrm{Sp}(4)\). It is concluded that the coordinate ring of the cone of \(\mathrm{O}(3)\) is not Koszul, hence the vanishing ideal of this cone has no quadratic Gr¨obner basis (although it is minimally generated by quadratic elements).

While recent years have witnessed a fast growth in mathematical artificial intelligence (AI). One of the most successful mathematical AI approaches is topological data analysis via persistent homology (PH) that provides explainable AI by extracting multiscale structural features from complex datasets. Interpretability is crucial for world models, the new frontier in AI that can understand and simulate reality. This article investigates the interpretability and representability of three foundational mathematical AI methods, PH, persistent Laplacians (PL) derived from topological spectral theory, and persistent commutative algebra (PCA) rooted in Stanley-Reisner theory. We apply these methods to a set of data, including geometric shapes, synthetic complexes, fullerene structures, and biomolecular systems to examine their geometric, topological, and algebraic properties. PH captures topological invariants such as connected components, loops, and voids through persistence barcodes. PL extends PH by incorporating spectral information, quantifying topological invariants, geometric stiffness, and connectivity via harmonic and nonharmonic spectra. PCA introduces algebraic invariants such as graded Betti numbers, facet persistence, and -vectors, offering combinatorial, topological, geometric, and algebraic perspectives on data over scales. Comparative analysis reveals that while PH offers computational efficiency and intuitive visualization, PL provides enhanced geometric sensitivity, and PCA delivers rich algebraic interpretability. Together, these methods form a hierarchy of mathematical representations, enabling explainable and generalizable AI for real-world data.

Let [Formula: see text] be a braided tensor category and [Formula: see text] a tensor category equipped with a braided tensor functor [Formula: see text]. For any exact indecomposable [Formula: see text]-module category [Formula: see text], we explicitly construct a right adjoint of the action functor [Formula: see text] afforded by [Formula: see text]. Here, [Formula: see text] is Müger’s centralizer of the subcategory [Formula: see text] inside the center [Formula: see text], also known as the relative center [20, 22]. The construction is parallel to the one presented by Shimizu [31], but using the relative (co)end [4] rather than the usual (co)end. This adjunction is monadic, and thus for the Hopf monad [Formula: see text], associated to it, there is a monoidal equivalence [Formula: see text] If [Formula: see text] is the right adjoint of [Formula: see text] then [Formula: see text] is the braided commutative algebra constructed in [21]. As a consequence of our construction of these algebras, in terms of the right adjoint to [Formula: see text], we can provide a recipe to compute them when [Formula: see text] is the category of finite-dimensional representations of a finite-dimensional Hopf algebra [Formula: see text] obtained by bosonization, and choosing an arbitrary [Formula: see text]-module category [Formula: see text]. We show an explicit example in the case of Taft algebras.

Abstract Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define a notion of generator complexity for commutative Frobenius algebras with combinatorial bases. In the context of finite groups, this gives rise to row and column versions of generator complexity for character tables. We investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.

In this paper, we examine a time-dependent family of two-dimensional algebras. We investigate the conditions under which any two algebras from this family, formed at different times, are isomorphic. Our findings reveal that the flow comprises of uncountable pairwise non-isomorphic algebras, including one commutative algebra. Additionally, we compare our results with a previously established classification of two-dimensional real algebras.

Background In the present paper, assuming every module H is unitary and every ring S is commutative with identity, we address the concepts of bounded modules and Endo-R.B.modules. We provide some examples, corollaries, remarks, and properties our new concept. An Endo-R.B S-module is considered to be more powerful than bounded module and Endo-R.B module that will present both of them in some details in the introduction. Motivated by these notions, we introduce and discuss a new class of modules called the Endo-Greatly Bounded module (abbreviated as Endo-G.B. module), a concept which has not been previously reported in the literature. Methods We formally define the Endo-G.B. module and provide examples, corollaries, and properties to illustrate this new concept. The study employs scalar modules and fully invariant submodules as crucial tools to establish connections. It must be emphasized that scalar modules and prime played a major role in achieving new results in this paper. We specifically investigate the important relationships between an Endo-G.B module and prime module as well as their generalizations. Results We get and prove that every Endo-G.B. module is Bounded, where as the converse is not generally true. Furthermore, we establish new results connecting our new class of S-modules to cyclic bounded modules, projective modules, multiplication modules, and finitely annihilated modules. We also derive necessary and sufficient conditions for an Endo-G.B. module to exhibit specific flexible properties. Conclusions In this paper, we presented a new class of S- modules called Endo-greatly bounded modules, these findings highlight the depth and flexibility of Endo-G.B. modules with in modern algebraic frameworks. This study opens new directions for further research in module theory, particularly in exploring the interaction between bounded ness and endomorphism structures, and may motivate future algebraic investigations in commutative algebra.

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.

Second-order derivative information, including mixed curvature, is central to Newton and trust-region optimization, uncertainty quantification, and simulation-based design. Classical finite differences (FD) remain popular but require delicate step-size tuning and can suffer from cancelation and noise amplification. Complex-step differentiation offers machine-precision gradients without subtractive cancelation, yet many second-derivative complex-step formulas reintroduce differencing. Hyper-dual numbers provide an algebraically principled alternative: by lifting real code to a four-component commutative nilpotent algebra, one obtains exact first and mixed second derivatives from a single evaluation, without finite differencing. This article gives a consolidated theoretical and experimental foundation for hyper-dual numbers. We formalize the algebra, prove exact Taylor truncation at second order, derive coefficient–extraction formulas for gradients and Hessians, and state assumptions for exactness, including limitations at non-smooth points and the need to branch on real parts. We present implementation patterns and language skeletons (C++, Python 3.11.5, MATLAB R2023b), and we provide fair numerical comparisons with FD, complex-step, and AD baselines. Stability tests under additive noise and ill-conditioning, together with runtime and memory profiling, demonstrate that hyper-dual coefficients are robust and reproducible in floating-point arithmetic, particularly for black-box codes where Hessian information is needed but differencing is fragile.

Commutativity of operator algebras