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Following a suggestion in a footnote of the paper [1] by Fano, we give a direct proof of the fact that one of the two Fano threefolds of degree 72 in P 38 is isomorphic to the anticanonically embedded weighted projective space P (1 , 1 , 4 , 6) . [ABSTRACT FROM AUTHOR]

This article is intended to indicate and discuss some errors in N. Lavi’s paper “A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions”. [ABSTRACT FROM AUTHOR]

We analyze the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebraHon thenth Weyl algebraA = An(K) over a fieldKof characteristic 0 factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebraHon an iterated Ore extension of derivation typein characteristic zero factors through a group algebra. [ABSTRACT FROM AUTHOR]

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let W be a classical Weyl group of rank n, with W ˜ its corresponding affine Weyl group. Our main result is that if X is a conjugacy class of involutions in W ˜ , then the commuting involution graph C (W ˜ , X) is either disconnected or has diameter at most n + 2. This bound is known to hold for types A ˜ n and C ˜ n , so the main work of this paper is to prove the theorem for types B ˜ n and D ˜ n. [ABSTRACT FROM AUTHOR]

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well. [ABSTRACT FROM AUTHOR]

AbstractFind a primary irreducible decomposition of an ideal is a problem which goes back to Emmy Noether. Find the irreducible decomposition of the powers of an ideal is a more recent problem. In our previous paper we have described the irreducible decomposition of the powers of the edge ideal IG of a simple, connected non bipartite graph G. In this paper, we use some tools from graph theory, mainly Whitney’s theorem on ear decompositions of 2-edge connected graphs in order to introduce a new method to make a graph factor-critical. We can describe the set ∪k=1∞Ass(IGk) in terms of some subsets of G. We give explicit formulas for the numbers astab(IG) and dstab(IG) which are the smallest number k such that Ass(IGk)=Ass(IGk+i) for all i≥0 and the smallest number k such that the maximal ideal belongs to Ass(IGk), respectively. We also give very simple upper bounds for astab(IG) and dstab(IG). [ABSTRACT FROM AUTHOR]

AbstractThe paper discusses perspective Abelian groups, defined as groups where isomorphic summands have a common complement. This concept is derived from the broader context of perspective R-modules. The paper explores various classes of these groups, providing characterizations and examples. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we prove that the variety Cm(g) of commuting m-tuples of elements of a simple Lie algebra g is usually reducible. More precisely, one of the main results of this paper shows that if m=3 and g is of types B,C,E7,E8,F4,G2 or Dl with l≥10, then Cm(g) is reducible. Moreover, if m≥4 and g is a simple Lie algebra different from sl2,sl3, then Cm(g) is reducible. In order to obtain these results, we need a more general result, which is another main result of this paper. It states that, under some mild assumptions, Cm(g) is reducible whenever Cm(l) is reducible for a Levi subalgebra l of g. [ABSTRACT FROM AUTHOR]

AbstractDerivations are linear operators which satisfy the Leibniz rule, while integrations are linear operators which satisfy the Rota-Baxter rule. In this paper, we introduce the notion of an FTC-pair, which consists of an algebra and module with a derivation and integration between them which together satisfy analogues of the two Fundamental Theorems of Calculus. In the special case of when an algebra is seen as a module over itself, we show that this sort of FTC-pair is precisely the same thing as an integro-differential algebra. We provide various constructions of FTC-pairs, as well as numerous examples of FTC-pairs including some based on polynomials, smooth functions, Hurwitz series, and shuffle algebras. Moreover, it is well known that integrations are closely related to Zinbiel algebras. The main result of this paper is that the category of FTC-pairs is equivalent to the category of Zinbiel algebras. [ABSTRACT FROM AUTHOR]

In a recent paper, Baker and Oliver Lorscheid showed that Descartes's Rule of Signs and Newton's Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebraic structures, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property. [ABSTRACT FROM AUTHOR]

In a finite group G, ρ (G) denotes the product of element orders of G. Let p , q , r , s and t be prime numbers. Recently, it was proved that C p 3 , Cpq, C p 2 q and Cpqrs are characterizable by the product of element orders, and if ρ (G) = ρ (C pqr) , then G ≅ A 5 or C pqr . In this paper, we continue this work and we show that C p × A 5 (for p > 5) and Cpqrst are characterizable by the product of element orders. In the rest of the paper, we show that if ρ (G) = ρ (C pqr) s ρ (C s) p 2 q , then (p , q , r , s) = (2 , 3 , 5 , 11) and G ≅ L 2 (11) . This shows that the structure of ρ (G) uniquely determined L 2 (11) and so L 2 (11) is the only group with this form of ρ (G) . [ABSTRACT FROM AUTHOR]

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : (N μ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q1 and Q2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form (α , α , γ) . [ABSTRACT FROM AUTHOR]

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. [ABSTRACT FROM AUTHOR]

This paper is concerned with computing the number of orbits under the action of the algebraic group E = E 6 (K) on the set of all pairs of root subgroups of itself. Furthermore, the number of orbits of the action of the group generated by a pair of certain reflections, on the 27 dimensional module A, has been computed. For the computations done in this paper, an explicit representation of E inside GL (A) has been fixed right at the onset. [ABSTRACT FROM AUTHOR]

This paper extends the Isaacs property, originally introduced by the second author, to noncommutative fusion rings through the use of a Fourier transform. It aligns this property with a subsequent categorical equivalent for spherical fusion categories described by Etingof et al., particularly in the pseudounitary case. Within the commutative framework, the Isaacs property is situated strictly between the integrality of structure constants and the 1- Frobenius properties. Furthermore, the paper demonstrates that the Extended Haagerup fusion categories (denoted as E H i ) do not possess the Isaacs property, thus providing a negative answer to questions posed by the aforementioned researchers. It also reaffirms that E H 1 lacks a braiding structure. [ABSTRACT FROM AUTHOR]

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the Jackson algebra, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a "non-commutative family" of central simple algebras and thus parameterizes Brauer classes over extensions of the base. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, S ⊆ R be a multiplicative set, and M be an R-module. We say that a submodule N of M with (N : R M) ∩ S = ∅ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules. [ABSTRACT FROM AUTHOR]

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms. [ABSTRACT FROM AUTHOR]

Let be a proper closed subset in the projective n-space over a field k of characteristic zero and let C(Y) be its affine cone. Let be the homogeneous defining ideal of Y with homogeneous maximal ideal. In this paper, we find out a lower bound of de Rham depth of C(Y) in terms of. [ABSTRACT FROM AUTHOR]

In this paper, we progress in the general study of rational functional identities in the setting of division rings. The main theorem in the paper states that the only additive maps f and g satisfying that f (x) = − x n g (x − 1) , for n = 3 or 4, are the zero maps. In addition, we provide an alternative proof to the result proved by Dar and Jing regarding the classification of additive maps f and g satisfying that f (x) = − x 2 g (x − 1) . [ABSTRACT FROM AUTHOR]

In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras. [ABSTRACT FROM AUTHOR]

In a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

In the present paper corrected versions of the statements in the paper "Description of finite-dimensional inner Rickart and Baer Jordan algebras" by F.N. Arzikulov and U.I. Khakimov are given. In particular, it is shown that for any Jordan algebra J with an idempotent p and an associative degenerate radical D such that J = F p + ̇ D , J is an inner RJ-algebra if and only if, for any nonzero a ∈ D , a 2 = 0 and p(pa) = pa. Also, other equivalent conditions when a Jordan algebra J is an inner RJ-algebra are given. As for finite-dimensional nilpotent Jordan algebras, there is not a nilpotent inner RJ-algebra (and hence inner BJ-algebra) except the finite-dimensional Jordan algebra the square of each element of which is zero. [ABSTRACT FROM AUTHOR]

In this paper, we mainly research Rota-Baxter co-operators on commutative Hopf algebras. We get a new Hopf algebra and some Hopf co-braces via Rota-Baxter co-operators on commutative Hopf algebras. Finally, we generalize Rota-Baxter co-operators to relative Rota-Baxter co-operators of Hopf algebras, construct a Hopf cobrace by a relative Rota-Baxter co-operator of commutative Hopf algebra, and establish the relation of relative Rota-Baxter co-operators and bijective 1-cocycles. [ABSTRACT FROM AUTHOR]

In this paper, we first give some constructions of right (resp. left) Leibniz H-pseudoalgebras and explore the characterization of symmetric Leibniz H-pseudoalgebras. Furthermore, we introduce the notion of relative Rota-Baxter H-operator and show that a right (resp. left) Leibniz H-pseudoalgebra together with a relative Rota-Baxter H-operator gives rise to a representation of related right (resp. left) Leibniz H-pseudoalgebra. Finally, we describe the central extension and the formal deformation in terms of the corresponding second cohomology group. [ABSTRACT FROM AUTHOR]

Given a finite group G, denote by cs(G) the set of the sizes of the conjugacy classes of G and by Cent(G) the set of the centralizers of elements of G. Consider a prime p and integers s ≥ 2 and n ≥ 2 , with gcd (p , s) = 1 . In this paper, some relations between cs(G) and | Cent (G) | are established in the case where cs (G) = { 1 , p n , p n − 1 s } . Further, when p ∈ { 2 , 3 } , we determine the values of s and the structure of a finite group G such that cs (G) = { 1 , p n , p n − 1 s } . We also describe the structure of an AC-group G such that [ G : Z (G) ] = 3 n s and | Cent (G) | = 1 + ∑ i = 0 n 3 i , where gcd (3 , s) = 1 . [ABSTRACT FROM AUTHOR]

Let φ : R → S be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between R and S are investigated. For an Artinian R-module A, the structure of A ⊗ R S is described. As an application, the Artinianess of certain local cohomology modules is clarified. [ABSTRACT FROM AUTHOR]

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras. [ABSTRACT FROM AUTHOR]

A way to construct the natural representation of the quantized affine algebra U v ( s l ̂ l) is via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for U q (s l N) were q is a root of unity to the action of U v ( s l ̂ l) as N tends toward infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional U q (g) -modules into Fock spaces of different charges and define an action of an affine quantum symmetric pair that plays the role of the quantized affine algebra. We describe how the action is related to the linkage principal for quantum groups at a root of unity and tensor product multiplicities. [ABSTRACT FROM AUTHOR]

Let D be a domain with its quotient field K. Let K [ θ ] = K [ x ] / (x n) where x is an indeterminate and θ = x + (x n) . Denote by R n = D + θ K [ θ ] , which is a subring of K [ θ ] . This paper studies several properties of Rn: coherence, weakly factorial rings, strong Mori rings, rings with Noetherian spectrum, weak normality, and t-closedness. [ABSTRACT FROM AUTHOR]

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n (K) (T n (D)) : = { f ∈ T n (K) [ x ] | f (T n (D)) ⊆ T n (D) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n (R 2) (Ω , T n (R 1)) : = { f ∈ T n (R 2) [ x ] | f (Ω) ⊆ T n (R 1) } for some subsets Ω ⊆ T n (R 1) . We generalize Frisch's result and show that Int T n (R 2) (T n (R 1)) : = Int T n (R 2) (T n (R 1) , T n (R 1)) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings. [ABSTRACT FROM AUTHOR]

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216. [ABSTRACT FROM AUTHOR]

In this paper, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a characterization of prime partial skew groupoid rings, and in particular of prime groupoid rings, thereby generalizing a classical result by Connell and partially generalizing recent results by Steinberg. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of NN-torsion-free modules (respectively, NN-injective modules) as a generalization of nonnil-torsion-free modules (respectively, injective modules). We show that they behave in a way similar to the classical ones. As an application, we characterize ϕ-rings over which every NN-torsion-free module is NN-injective. [ABSTRACT FROM AUTHOR]

In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p (x 1 , ... , x m) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1 ≤ r ≤ n − 1 , where r is the order of p. Then p (T n (K)) is a dense subset of T n (K) (r − 1) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023. [ABSTRACT FROM AUTHOR]

In this paper, we provide a local method to find compositional inverses of all PPs, some new PPs and their compositional inverses are given. [ABSTRACT FROM AUTHOR]

Which groups can occur as the group of units in a ring? Such groups are called realizable. Though the realizable members of several classes of groups have been determined (e.g., cyclic, odd order, alternating, symmetric, finite simple, indecomposable abelian, and dihedral), the question remains open. The general linear groups are realizable by definition: they are the units in the corresponding matrix rings. In this paper, we study the realizability of two closely related linear groups, the special linear groups and the affine general linear groups. We determine which special linear groups of degree 2 over a finite field are realizable by a finite ring, and we determine which affine general linear groups of degree 1 over a cyclic group are realizable by a finite ring. We also give partial results for certain linear groups of other degrees and for rings of characteristic zero. [ABSTRACT FROM AUTHOR]

A ring extension is a ring homomorphism preserving identities. In this paper, we give the definitions of relative injective envelopes and relative projective covers of modules on ring extensions, and study their basic properties. In particular, we give their equivalent characterizations in terms of relative essential monomorphisms and relative superfluous epimorphisms, and prove that relative injective envelopes and relative projective covers on ring extensions are unique up to isomorphism whenever they exist. Moreover, for an extension of Artin algebras, we show that every finitely generated module has both a relative injective envelope and a relative projective cover. In addition, we compare relative injective envelopes and relative projective covers on two ring extensions linked by surjective homomorphisms of rings respectively. [ABSTRACT FROM AUTHOR]

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point t 0 ∈ P 1 (K) of some finite regular Galois extension F / K (T) with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the l -universal Frattini cover of an arbitrary finite group. [ABSTRACT FROM AUTHOR]

We study the rings Nil D of nilpotent elements of R-derivations D on polynomial rings over a UFD R containing Q . The main result of this paper determines Nil D of the monomial derivations D on R [ x , y ] . Furthermore, as its application, we prove that, for each positive integer m, there exists a monomial R-derivation with a slice on R [ x , y , z ] such that the minimal number of generators of its kernel equals m. [ABSTRACT FROM AUTHOR]