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AbstractFollowing 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 P38 is isomorphic to the anticanonically embedded weighted projective space P(1,1,4,6). [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]

AbstractThe 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]

AbstractIn 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=Fp+̇D, J is an inner RJ-algebra if and only if, for any nonzero a∈D, a2=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]

AbstractIn 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 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]

This paper deals with additive groups of rings in which all subrings are ideals. It is shown that if an abelian group supports only rings with this property, then all of them are commutative. This result is obtained for associative as well as not necessarily associative rings. Some previously known far from obvious results related to the mentioned groups are generalized and complemented. In particular, the classification of torsion-free abelian groups on which every not necessarily associative ring has the property that all its subrings are ideals is provided up to the structure of nil-groups. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a kind of decomposition of a finite group called a uniform group factorization, as a generalization of exact factorizations of a finite group. A group G is said to admit a uniform group factorization if there exist subgroups H 1 , H 2 , ... , H k such that G = H 1 H 2 ⋯ H k and the number of ways to represent any element g ∈ G as g = h 1 h 2 ⋯ h k ( h i ∈ H i ) does not depend on the choice of g. Moreover, a uniform group factorization consisting of cyclic subgroups is called a uniform cyclic group factorization. First, we show that any finite solvable group admits a uniform cyclic group factorization. Second, we show that whether all finite groups admit uniform cyclic group factorizations or not is equivalent to whether all finite simple groups admit uniform group factorizations or not. Lastly, we give some concrete examples of such factorizations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 × 2 matrices over an algebraically closed field of characteristic different from 2, 3 and 5 in the class of the commutative power-associative algebras. All irreducible non-unital modules, and irreducible unital modules up to dimension three are classified, namely we find seven non-parameterized and five families of parameterized modules of dimension three. For every k ≥ 2 , an irreducible module of dimension 3k is also constructed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some constructions results and examples of n-BiHom-Lie color algebras. Next, we introduce the definition of BiHom-modules over n-BiHom-Lie color algebras and we provide some properties. Moreover we investigate generalized derivations of n-BiHom-Lie color algebras and their BiHom-subalgebras. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the positivity of double point divisors of general inner projections for projective S2-varieties with invertible dualizing sheaves, without assuming the smoothness or the normality of varieties. As applications, we characterize two types of projective varieties: varieties with trivial double point divisors and varieties with d − e − 1 ≤ n for dimension n, degree d and codimension e. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. An R-module M is said to be w-split if Ext R 1 (M , N) is a GV-torsion R-module for all R-modules N. It is known that every projective module is w-split, but the converse is not true in general. In this paper, we study the w-split dimension of a flat module. To do so, we introduce and study the so-called w-copure (resp., strongly w-copure) projective modules, which is in some way a generalization of the notion of copure (resp., strongly copure) projective modules. An R-module M is said to be w-copure projective (resp., strongly w-copure projective) if Ext R 1 (M , N) (resp., Ext R n (M , N) ) is a GV-torsion R-module for all flat R-modules N and any n ≥ 1 . [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and Nil(R) denote the set of nilpotent elements of R. In this paper, we investigate two generalizations of exact sequences, ϕ -exact sequences and N -exact sequences. With the help of two generalizations of exact sequences, we generalize the concept of homology modules. We show that they behave in a way similar to the classical ones. [ABSTRACT FROM AUTHOR]

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ (n) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ. [ABSTRACT FROM AUTHOR]

Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that G ≅ f (H) , and completely f-realisable if there is a group H such that G ≅ f (H) and every subgroup of G is isomorphic to f (H 1) for some subgroup H 1 of H and vice versa. Denote by L (G) the absolute center of a group G, that is the set of elements of G fixed by all automorphisms of G. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups C N , N ∈ N * , are completely L-realisable. [ABSTRACT FROM AUTHOR]

This paper describes properties of three certain classes of modules M over a ring R determined by conditions on isomorphic direct summands ( ≲ ⊕ ): The condition that whenever (I m λ ≅) M / Ker λ ≲ ⊕ M then Ker λ and I m λ are direct summands of M for any endomorphism λ ∈ End (M) (kernel-endoregular modules). The condition that if M / A ≅ B where A, B ≲ ⊕ M then M / B ≅ A (iso-summand-morphic modules). The condition if M / A ≅ B where A, B ≤ ⊕ M , then M / B ≅ A (summand-morphic modules) which is precisely the internal cancellation property for modules. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new approach to construct monoidal Hom-Hopf algebras. We investigate monoidal Hom-Hopf algebra structure on a left (n, l)-Hom-crossed coproduct structure with a left n-Hom-smash product structure, obtaining Radford [ n , (n , l) ] -biproduct structure theorem. Then, we study a Hom-coaction admissible mapping system to characterize this Radford [ n , (n , l) ] -biproduct structure. Finally, we study the cosemisimplicity of a special Hom-smash coproduct and prove the related Maschke theorem. [ABSTRACT FROM AUTHOR]

In 1994, Long and Moody introduced a method to construct a new representation of the braid group from the representation of the braid group or the semidirect product of the braid group and the free group. In this paper, we show that its matrix presentation is written using the Fox derivative, and also a relation with twisted Alexander invariants. Communicated by Eric Jespers [ABSTRACT FROM AUTHOR]

Let Γ be a torsion-free cancellative monoid and let R = ⊕ α ∈ Γ R α be a graded integral domain. In this paper, we show that if R is a graded Strong Mori domain, then every homogeneously t-linked overring of R contained in R hwg is a graded Strong Mori domain, where Rhwg is the homogeneous w-global transform of R; and if R is a graded Strong Mori domain, then R S is also a graded Strong Mori domain, where S is a multiplicative set of ideals of R generated by homogeneous ideals of R. [ABSTRACT FROM AUTHOR]

Let G be a generalized matrix algebra. A linear map ϕ : G → G is said to be a left (right) Lie centralizer at E ∈ G if ϕ ([ S , T ]) = [ ϕ (S) , T ] ( ϕ ([ S , T ]) = [ S , ϕ (T) ] ) holds for all S , T ∈ G with ST = E. ϕ is of a standard form if ϕ (A) = Z A + γ (A) for all A ∈ G , where Z is in the center of G and γ is a linear map from G into its center vanishing on each commutator [ S , T ] whenever ST = E. In this paper, we give a complete characterization of ϕ. It is shown that, under some suitable assumptions on G , ϕ has a standard form. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q. In this paper, we investigate the permutation property of the quadrinomial f (x) = x + a 1 x q (q − 1) + 1 + a 2 x q + a 3 x 2 (q − 1) + 1 over F q 2 , where q is odd and a 1 , a 2 , a 3 ∈ F q * . More precisely, we obtain the necessary and sufficient conditions for f (x) to be a permutation polynomial over F q 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study the class of affine semigroups generated by integral vectors, whose components are in generalized arithmetic progression and we observe that the defining ideal is determinantal. We also give a sufficient condition on the defining ideal of the semigroup ring for the equality of the Betti numbers of the defining ideal and those of its initial ideal. We introduce the notion of an affine semigroup generated by join of two affine semigroups and show that it preserves some nice properties, including Cohen-Macaulayness, when the constituent semigroups have those properties. [ABSTRACT FROM AUTHOR]

Let F be a non-cyclic free group of rank n. Consider the quotient F / [ F ″ , F ] , the free centre-by-metabelian group of rank n. In 1973, C. K. Gupta proved by purely group theoretic means that it contains an elementary abelian 2-group of rank ( 4 n) in its centre for n ≥ 4 , and exhibited an explicit generating set for this torsion subgroup. In this paper, using homological methods, we provide an alternative explicit generating set for it, and identify this torsion subgroup as the isolator of an explicitly given subgroup in the quotient F ″ / [ F ″ , F ′ ] . [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of the existence of a primitive element α ∈ F q such that f (α) is also primitive in F q for a given non-exceptional f (x) ∈ F q (x) with its degree sum greater than 1. We obtain a condition on q for the existence of such primitive elements when q ≡ 3 (mod 4) and f(x) belongs to the family of even or odd functions. [ABSTRACT FROM AUTHOR]

In 2020, Cossu and Zanardo raised a problem on the idempotent factorization of singular matrices in the form ( p z z ¯ ‖ z ‖ / p ) , where p is a prime integer which is irreducible but not prime in the ring of integers O K , with K a real quadratic number field, and z ∈ O K is such that 〈 p , z 〉 is a non-principal ideal. In this paper, we provide some classes of matrices that factor in a prescribed way and some other classes that do not. We further show that there are matrices in the above form that cannot be written as a product of two idempotent matrices. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to calculate the index of any number field K generated by a root of an irreducible trinomial x 4 + a x + b ∈ Z [ x ] . Our approach is based on Engstrom's results and the factorization of 2 Z K and 3 Z K . In particular, we reformulate the Davis and Spearman's results. Namely, for every prime integer p, we evaluate ν p (i (K)) . The existence of a common index divisor of K guaranties the non monogenity of K. [ABSTRACT FROM AUTHOR]

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ (n) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ. [ABSTRACT FROM AUTHOR]

AbstractLet p be a prime and Fp be a finite field of p elements. Let FpG denote the group algebra of the finite p-group G over the field Fp and V(FpG) denote the group of normalized units in FpG. Suppose that G is a finite p-group given by a central extension of the form 1→Zpn×Zpm→G→Zp×⋯×Zp→1 and G′≅Zp, n,m≥1 and p is odd. In this paper, the structure of G is determined. And the relations of V(FpG)pl and Gpl, Ωl(V(FpG)) and Ωl(G) are given. Furthermore, there is a direct proof for V(FpG)p∩G=Gp. [ABSTRACT FROM AUTHOR]

AbstractLet a be an ideal of a given commutative Noetherian ring R which satisfies the condition of the first problem of R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145–164]. In this paper, we prove that a also satisfies the condition of his second problem in the same article. We also provide an example to show that the converse statement does not hold in general. [ABSTRACT FROM AUTHOR]

AbstractThe notion of double extension for symplectic Lie algebras was introduced by Medina and Revoy in 1985. In 2021, Valencia gave a condition for symplectic Lie algebras which are obtained through double extension to admit compatible left-symmetric structures. In this paper, we formulate the notion of double extension for left-symmetric structures and give a condition for left-symmetric structures to be obtained through double extension. Moreover, we prove that right nilpotent left-symmetric structures on some solvable Lie algebras are obtained through double extension. We also give a condition for left-symmetric structures which are obtained through double extension to be complete. [ABSTRACT FROM AUTHOR]

AbstractFor an irreducible character χ of a finite group G, the codegree of χ is defined as |G:ker(χ)|/χ(1). In this paper, we characterize finite p-groups with three character codegrees. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, Whittaker modules are studied for a subalgebra qϵ of the N =2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of the Whittaker modules are given. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we study the modular structure of the Foulkes module H(2m) of the symmetric group S2m acting on set partitions of a set of size 2m into m sets each of size 2, defined over a field of positive characteristic p. We aim to understand the decomposition of the tensor powers of H(2m). In particular, we study the asymptotics of the non-projective part of tensor powers of these modules by computing the gamma invariant as defined by Dave Benson and Peter Symonds. [ABSTRACT FROM AUTHOR]

AbstractSince the existing tropical cryptographic protocols are either susceptible to the Kotov-Ushakov attack and its generalization, or to attacks based on tropical matrix periodicity and predictive behavior, several attempts have been made to propose protocols that resist such attacks. Despite these attempts, many of the proposed protocols remain vulnerable to attacks targeting the underlying hidden problems, one of which we call the tropical two-sided discrete logarithm with shift. An illustrative case is the tropical Stickel protocol, which, when formulated with a single monomial instead of a polynomial, becomes susceptible to attacks based on solutions of the above mentioned tropical version of discrete logarithm. In this paper we will formally introduce the tropical two-sided discrete logarithm with shift, discuss how it is solved, and subsequently demonstrate an attack on a key exchange protocol based on the tropical semiring of pairs. This particular protocol is compromised due to the existence of efficient (albeit heuristic) solution of the tropical two-sided logarithm problem, and this highlights the ongoing challenges in search of a “good” key exchange protocol in tropical cryptography. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we first investigate some properties of relative Rota-Baxter operators on Leibniz conformal algebras with respect to representations and their connections with Leibniz dendriform conformal algebras. Next, we introduce the notion of a twisted relative Rota-Baxter operator and construct a conformal NS-Leibniz algebra structure related to twisted relative Rota-Baxter operators. Furthermore, we define the cohomology of a twisted relative Rota-Baxter operator with coefficients in a suitable representation. Finally, we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we define a partially capable Lie superalgebra. As an application, we classify all capable nilpotent Lie superalgebras of dimension less than or equal to five. [ABSTRACT FROM AUTHOR]

AbstractLet D be an F-central division algebra. In this paper, we investigated the exponent of the group G(D)=D*/NrdD(D*)D′, where D* is the group of units of D, NrdD(D*) is the image of D* under the reduced norm map and D′ is the commutator subgroup of D*. We show that if  exp (G(D))D and F satisfy strong conditions. In particular, we observe that if D is a sum cyclic algebras in Br(F), then  exp (G(D))F is euclidean and D is a tensor product of an ordinary quaternion algebra and a division algebra of odd index. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we study the relationship among three n-cluster tilting subcategories of triangulated categories in a recollement. Let (D′,D,D″) be a recollement of triangulated categories. We construct an n-cluster tilting subcategory of D by gluing together n-cluster tilting subcategories of D′ and D″ under certain conditions. On the other hand, we prove that an n-cluster tilting subcategory of D can induce n-cluster tilting subcategories of D′ and D″. Using these two results, we give a one-to-one correspondence between some n-cluster tilting subcategories of D and pairs of n-cluster tilting subcategories of D′ and D″. Furthermore, we obtain a recollement of abelian categories from a recollement of triangulated categories. [ABSTRACT FROM AUTHOR]

AbstractLet A be a ring and λ:R≥0→R≥0 be an increasing nonzero function. In this paper, we show that if A is a Noetherian ring with characteristic n≠0 and limk→+∞λ(k+1)λ(k)=+∞, then A[X,Y;λ] is an SFT ring. This result allows us to construct a nonNoetherian SFT ring which has some Noetherian completion. Also, we use the composite ring extension to construct many examples of such rings. Many examples are provided. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we study the Grothendieck groups of m-cluster categories of type Dn, denoted by CDnm, when m is odd, via the m-cluster tilting subcategories of CDnm. When m = 1, we generalize the work of Barot, Kussin and Lenzing for studying the Grothendieck groups of cluster categories of type Dn. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we introduce proximal groups that are a generalization of topological groups. A straight-forward application of proximal groups is given in the concise representation of collections of overlapping Hilbert envelope lobes attached to the peak points on oscillatory motion waveforms that occur in sequences of video frames that track object movements. [ABSTRACT FROM AUTHOR]

AbstractLet R be an associative unital ring, and let f∈R[x]. We say that f is a left annihilating content (AC) polynomial if f = af1 for some a∈R and f1∈R[x] with lR[x](f1)=0. The ring R is called a left EM-ring if each f∈R[x] is a left AC polynomial. In this paper, it is shown that R is a left EM-ring if and only if R is a left McCoy ring, and for each finitely generated right ideal I of R, there is an element a∈R and a finitely generated right ideal J of R with lR(J)=0 and I = aJ. If R is a left duo right Bezout ring, then R is a left EM-ring and has property (A). For a unique product monoid G, we show that if R is a reversible left EM-ring, then the monoid ring R[G] is also a left EM-ring. Additionally, for a reversible right Noetherian ring R, we prove that R, R[x],R[x,x−1], and R[[x]] are all simultaneously left EM-rings. Finally, we give an application of left EM-rings (resp. strongly left EM-rings) in studying the graph of zero-divisors of polynomial rings (resp. power series rings). [ABSTRACT FROM AUTHOR]

AbstractLet F be a saturated fusion system over a p-group S and let E be a subsystem of F with AutE(S)=AutF(S). We prove that the fusion system F is generated by all E-morphisms and all F-automorphisms of some domestic intersections (called F∖E-domestic intersections in the paper) of F. We also get some results about control of fusion in finite groups as applications. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the orders of odd-order composition factors in a composition series of an arbitrary finite linear group. [ABSTRACT FROM AUTHOR]

AbstractThe goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry. [ABSTRACT FROM AUTHOR]

AbstractThis paper aims to introduce the concept of nilpotency and capability in multiplicative Lie algebras. Also, we see the existence of covers of a multiplicative Lie algebra and thoroughly examine their relationships with capable and perfect multiplicative Lie algebras. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim[g,g]=1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l2n+1J1 , where J1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. [ABSTRACT FROM AUTHOR]