Commutative Algebra Research Articles

Derivations in disjointly complete commutative regular algebras

We show that any nonexpansive derivation on a subalgebra of a dis-jointly complete commutative regular algebra extends up to a derivation on . For an algebra of functions , continuous on a dense open subset of Stone compact X, we establish that the lack of nontrivial derivation is equivalent to σ-distributivity of the Boolean algebra of clopen subsets of X. The field is an arbitrary normed field of charachteristic zero containing a complete non-discrete subfield. Our work is motivated by two seemingly unrelated problems due to Ayupov [2] and Wickstead [32].

On Pre-Novikov Algebras and Derived Zinbiel Variety

For a non-associative algebra $A$ with a derivation $d$, its derived algebra $A^{(d)}$ is the same space equipped with new operations $a\succ b = d(a)b$, $a\prec b = ad(b)$, $a,b\in A$. Given a variety ${\rm Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in ${\rm Var}$ and for all derivations $d$ of $A$. The same terminology is applied to binary operads governing varieties of non-associative algebras. For example, the operad of Novikov algebras is the derived one for the operad of (associative) commutative algebras. We state a sufficient condition for every algebra from a derived variety to be embeddable into an appropriate differential algebra of the corresponding variety. We also find that for ${\rm Var} = {\rm Zinb}$, the variety of Zinbiel algebras, there exist algebras from the derived variety (which coincides with the class of pre-Novikov algebras) that cannot be embedded into a Zinbiel algebra with a derivation.

A CALCULATION OF THE PERFECTOIDIZATION OF SEMIPERFECTOID RINGS

AbstractWe show that perfectoidization can be (almost) calculated by using p-root closure in certain cases, including the semiperfectoid case. To do this, we focus on the universality of perfectoidizations and uniform completions, as well as the p-root closed property of integral perfectoid rings. Through this calculation, we establish a connection between a classical closure operation “p-root closure” used by Roberts in mixed characteristic commutative algebra and a more recent concept of “perfectoidization” introduced by Bhatt and Scholze in their theory of prismatic cohomology.

On the behavior of Massey products under field extension

We show that global vanishing of Massey products on a commutative differential graded algebra is not invariant under field extension. Non-vanishing triple Massey products remain non-vanishing upon field extension, while higher Massey products can generally vanish. If the field being extended is algebraically closed, all non-vanishing Massey products remain non-vanishing on a finite type commutative differential graded algebra.

Noncompact uniform universal approximation

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space Rn. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all activation functions φ that are continuous, nonpolynomial, and asymptotically polynomial at ±∞. When φ is moreover bounded, we exactly determine which functions can be uniformly approximated by neural networks, with the following unexpected results. Let Nφl(Rn)¯ denote the vector space of functions that are uniformly approximable by neural networks with l hidden layers and n inputs. For all n and all l≥2, Nφl(Rn)¯ turns out to be an algebra under the pointwise product. If the left limit of φ differs from its right limit (for instance, when φ is sigmoidal) the algebra Nφl(Rn)¯ (l≥2) is independent of φ and l, and equals the closed span of products of sigmoids composed with one-dimensional projections. If the left limit of φ equals its right limit, Nφl(Rn)¯ (l≥1) equals the (real part of the) commutative resolvent algebra, a C*-algebra which is used in mathematical approaches to quantum theory. In the latter case, the algebra is independent of l≥1, whereas in the former case Nφ2(Rn)¯ is strictly bigger than Nφ1(Rn)¯.

Local, colocal, and antilocal properties of modules and complexes over commutative rings

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles.

Rational Involutions and an Application to Planar Systems of ODE

An involution refers to a function that acts as its own inverse. In this paper, our focus lies on exploring two-dimensional involutive maps defined by rational functions. These functions have denominators represented by polynomials of degree one and numerators by polynomials of a degree of, at most, two, depending on parameters. We identify the sets in the parameter space of the maps that correspond to involutions. The investigation relies on leveraging algorithms from computational commutative algebra based on the Groebner basis theory. To expedite the computations, we employ modular arithmetic. Furthermore, we showcase how involution can serve as a valuable tool for identifying reversible and integrable systems within families of planar polynomial ordinary differential equations.

On the structure and classification of Bernstein algebras

Linear algebra tools are used to give a new approach to the open problem of the classification of Bernstein algebras. We prove that any Bernstein algebra [Formula: see text] is isomorphic to a semidirect product [Formula: see text] associated to a commutative algebra [Formula: see text] such that [Formula: see text], for all [Formula: see text] and an idempotent endomorphism [Formula: see text] of [Formula: see text] satisfying two compatibility conditions. The set of types of [Formula: see text]-dimensional Bernstein algebras is parametrized by an explicitly constructed classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups [Formula: see text].

Locally free representations of quivers over commutative Frobenius algebras

Locally free representations of quivers over commutative Frobenius algebras

Deformations of commutative Artinian algebras

The paper is devoted to the study of deformations of Artinian algebras and zero-dimensional germs of varieties. In particular, an approach is developed to solving the open problem about the nonexistence of rigid Artinian algebras; it is based essentially on the use of the canonical duality in the cotangent complex. Thus, it is shown that there are no rigid Gorenstein Artinian algebras and rigid almost complete intersections. The proof of the latter statement is based on the properties of the torsion functors. More precisely, the tensor product of the conormal and canonical modules of the corresponding Artinian algebra is calculated. In this case, the homology and cohomology groups of higher degrees are also found. Among other things, some estimates are obtained for the dimension of the spaces of the first lower and upper cotangent functors of Artinian algebras, and the relationship between them is described. In conclusion, several examples of nonsmoothable Artinian noncomplete intersections are examined, and some unusual properties of such algebras are discussed.

The finite dual coalgebra as a quantization of the maximal spectrum

In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A∘ act as an appropriate depiction of the noncommutative maximal spectrum of A; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces.

Isomorphism classes of Drinfeld modules over finite fields

We study isogeny classes of Drinfeld A-modules over finite fields k with commutative endomorphism algebra D, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order A[π] of D generated by the Frobenius π occurs as an endomorphism ring by proving when it is locally maximal at π, and show that this happens if and only if the isogeny class is ordinary or k is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring E of a Drinfeld module ϕ up to D-linear equivalence acts on the isomorphism classes in the isogeny class of ϕ, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases, which can be implemented as a computer algorithm.

Bases of fixed point subalgebras on nilpotent Leibniz algebras

Let K be a field of characteristic zero, X={x_(1,) x_2,…,x_n} and R_m={r_(1,) ,…,r_m} be two sets of variables, F be the free left nitpotent Leibniz algebra generated by X, and K[R_m ] be the commutative polynomial algebra generated by R_m over the base field K. The fixed point subalgebra of an automorphism φ is the subalgebra of F consisting of elements that are invariant under the automorphism. In this work, we consider specific automorphisms of F and determine the fixed point subalgebras of these automorphisms. Then, we find bases of these fixed point subalgebras. In addition, we get generators of these subalgebras as a free K[R_m ] -module.

Construction of free quasi-idempotent differential Rota-Baxter algebras by Gröbner-Shirshov bases

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

Numerical representation of topological real algebras

We show that the isomorphism provided by the Gelfand-Mazur theorem for a commutative pre-ordered real Banach algebra A with unit defines a numerical representation which is compatible with the order structure.

A Study on Some Topics in Algebra

This study provides a comprehensive examination of various topics in algebra, exploring its historical development, fundamental concepts, and diverse branches. The paper delves into elementary algebra, abstract algebra, linear algebra, and commutative algebra, offering detailed insights into each area. Additionally, it introduces a problem-analysis model to evaluate algebraic concepts. This exploration aims to highlight the significance of algebra in mathematical theory and its practical applications, underscoring its role as a foundational pillar in both academic research and applied sciences.

On the Structure of Some Types of Higher Derivations

In this paper we introduce the concepts of higher {Lgn , Rhn }-derivation, higher {gn, hn}-derivation and Jordan higher {gn, hn}-derivation. Then we give a characterization of higher {Lgn , Rhn }-derivations and higher {gn, hn}-derivations in terms of {Lg, Rh}-derivations and {g, h}-derivations, respectively. Using this result, we prove that every Jordan higher {gn, hn}-derivation on a semiprime algebra is a higher {gn, hn}-derivation. In addition, we show that every Jordan higher {gn, hn}- derivation of the tensor product of a semiprime algebra and a commutative algebra is a higher {gn, hn}-derivation. Moreover, we show that there is a one to one correspondence between the set of all higher {Lgn , Rhn }-derivations and the set of all sequences of {LGn , RHn }-derivations. Also, it is presented that if A is a unital algebra and {fn} is a generalized higher derivation associated with a sequence {dn} of linear mappings, then {dn} is a higher derivation. Some other related results are also discussed.

Algorithmic complexity for theories of commutative Kleene algebras

Kleene algebras are structures with addition, multiplication and constants $0$ and $1$, which form an idempotent semiring, and the Kleene iteration operation. In the particular case of $*$-continuous Kleene algebras, Kleene iteration is defined, in an infinitary way, as the supremum of powers of an element. We obtain results on algorithmic complexity for Horn theories (semantic entailment from finite sets of hypotheses) of commutative $*$-continuous Kleene algebras. Namely, $\Pi_1^1$-completeness for the Horn theory and $\Pi^0_2$-completeness for its fragment, where iteration cannot be used in hypotheses, is proved. These results are commutative counterparts of the corresponding theorems of D. Kozen (2002) for the general (non-commutative) case. Several accompanying results are also obtained.

Completeness of derivation algebras of finite commutative group algebras

Completeness of derivation algebras of finite commutative group algebras

On the structure of graded 3-Lie-Rinehart algebras

We study the structure of a graded 3-Lie-Rinehart algebraLover an associative and commutative graded algebra A. For G an abelian group, we show that if (L,A) is a tight G-graded 3-Lie-Rinehart algebra, then L and A decompose as L = ? i?I Li and A = ? j?J Aj, where any Li is a non-zero graded ideal of L satisfying [Li1 ,Li2 ,Li3] = 0 for any i1, i2, i3 ? I different from each other, and any Aj is a non-zero graded ideal of A satisfying AjAl = 0 for any l, j ? J such that j ?l, and both decompositions satisfy that for any i ? I there exists a unique j ? J such that AjLi ? 0. Furthermore, any (Li,Aj) is a graded 3-Lie-Rinehart algebra. Also, under certain conditions, it is shown that the above decompositions of L and A are by means of the family of their, respectively, graded simple ideals.