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This paper analyzes some of the fundamental results of the unit groups of the commutative modular group algebras. We introduce a more visible and concise form of a number of these results. We consider with a corresponding justification some essentially inexact and unproved results in some papers and for certain of them either we mark the corrections or we point out the corrections, which are founded in another papers. [ABSTRACT FROM AUTHOR]

It is known that the depth of every factor ring of C (X) module an ideal is at most 1. In this paper, we examine conditions under which the depth of factor rings of C (X) module closed ideals are either 0 or 1. Particularly, we show that the depth of factor ring C (X) / M A , A ⊆ X , is 0 (or equivalently this ring is classical i.e. its every element is unit or zerodivisor) if and only if A is an almost P -space completely separated from every zero-set disjoint from it. Using this, it has been confirmed that C (X) modulo the smallest z ∘ -ideal containing f ∈ C (X) is classical if and only if cl X int X Z (f) is an almost P -space completely separated from every zero-set disjoint from it. Also, it has been verified that X is a P -space if and only if for every ideal I ⊆ C (X) , the factor ring C (X) / I has depth zero. Finally, we present a counterexample to a conjecture about the depth of subrings of C (X) in the literature. [ABSTRACT FROM AUTHOR]

In the last decade, two new approaches have been introduced for the analysis of the projectivity of modules. In this paper, the projectivity of modules has been studied with a new approach by using projectively generated proper classes. As an opposite to projectivity, a module M is said to be π -indigent if the projectively generated proper class by M is consisting of exactly the split short exact sequences. We consider rings over which every (finitely generated, finitely presented, cyclic, simple) right R -modules are either projective or π -indigent. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [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. In this paper, we determine completely Aut -realisable groups. We also study f -realisable groups for f = Z , F , M , D , Φ , where Z (H) , F (H) , M (H) , D (H) and Φ (H) denote the center, the Fitting subgroup, the Chermak–Delgado subgroup, the derived subgroup and the Frattini subgroup of the group H , respectively. [ABSTRACT FROM AUTHOR]

This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree. [ABSTRACT FROM AUTHOR]

Assume that we are given a sequence F of k homogeneous polynomials in n variables of degree at most d and the ideal ℐ generated by this sequence. The aim of this paper is to present a new and effective method to determine, within the arithmetic complexity d O (n) , whether F is regular. This algorithm has been implemented in Maple and its efficiency (compared to the classical approaches for regular sequence test) is evaluated via a set of benchmark polynomials. Furthermore, we show that if F is regular then we can transform ℐ into Nœther position and at the same time compute a reduced Gröbner basis for the transformed ideal within the arithmetic complexity d O (n 2) . Finally, under the same assumption, we establish the new upper bound 2 (d k / 2) 2 n − k − 1 for the maximum degree of the elements of any reduced Gröbner basis of ℐ in the case that n > k. [ABSTRACT FROM AUTHOR]

We first recall the construction of a twisted pre-Lie algebra structure on the species of finite connected topological spaces. Then, we construct a corresponding non-coassociative permutative (NAP) coproduct on the subspecies of finite connected T 0 topological spaces, i.e. finite connected posets, and we prove that the vector space generated by isomorphism classes of finite posets is a free pre-Lie algebra and also a cofree NAP coalgebra. Furthermore, we give an explicit duality between the non-associative permutative product and the proposed NAP coproduct. Finally, we prove that the results in this paper remain true for finite connected topological spaces. [ABSTRACT FROM AUTHOR]

In this paper, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by Fonseca and Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution. [ABSTRACT FROM AUTHOR]

The SFT (for strong finite type) condition was introduced by [J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973) 299–304] in the context of studying the condition for formal power series rings to have finite Krull dimension. In the context of commutative rings, the SFT property is a near-Noetherian property that is necessary for a ring of formal power series to have finite Krull dimension behavior. Many others have studied this condition in the context of the dimension of formal power series rings. In this paper, we explore a specialization (and in some sense a more natural) variant of the SFT property that we dub as the VSFT (for very strong finite type) property. As is true of the SFT property, the VSFT property is a property of an ideal that may be extended to a global property of a commutative ring with identity. Any ideal (respectively, ring) that has the VSFT property has the SFT property. In this paper, we explore the interplay of the SFT property and the VSFT property. [ABSTRACT FROM AUTHOR]

The paper contains the classification of non-commutative torsion-free associative rings of rank two. Furthermore, torsion-free abelian groups of rank two supporting an associative, but not commutative ring structure are classified. [ABSTRACT FROM AUTHOR]

In this paper, we study algebraic properties of a family of certain polynomials arising from the functional equation for Dickson polynomials. We see that the roots and discriminants of those polynomials have very simple expressions, and each polynomial is cyclic. Further, we provide an irreducibility criterion analogous to the well-known criterion of Vahlen-Capelli. We finish the paper by showing that any cyclic extension of a certain field comes from a member of the family. [ABSTRACT FROM AUTHOR]

The paper deals with new specific constructions of indecomposable torsion-free abelian groups of rank two and nonzero rings on them. They illustrate purely theoretical results and complement quite rare examples obtained during the classical as well as recent research of additive groups of rings. The presented results concerning the homogeneous groups remain true for groups of any finite nonzero rank. Moreover, the paper contains a construction of a torsion-free indecomposable abelian group of an arbitrary finite rank greater than two supporting an associative, but not commutative ring, as well as a ring which is neither associative nor commutative. [ABSTRACT FROM AUTHOR]

Let q denote the finite field with q elements. In this paper, we investigate the trinomial f (x) = x 4 q + 1 + α x 5 q + x q + 4 over the finite field 5 2 k , where α ∈ 5 k * with k being a positive integer. We prove that the trinomial f (x) permutes 5 2 k if and only if α = − 1 and k is even. This work is a continuation of the previous work of Bai and Xia [A new class of permutation trinomials constructed from Niho exponents, Cryptogr. Commun. 10 (2018) 1023–1036]. [ABSTRACT FROM AUTHOR]

The cyclic graph Γ (S) of a semigroup S is the simple undirected graph whose vertex set is S and two vertices x , y are adjacent if the subsemigroup generated by x and y is monogenic. In this paper, we determine the clique number of Γ (S) for an arbitrary semigroup S. Further, we obtain the independence number of Γ (S) if S is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of Γ (S) if S is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds. [ABSTRACT FROM AUTHOR]

This paper is devoted to defining and studying Whittaker modules and high order Whittaker modules for the N = 1 super-BMS3 algebra. We also classify the simple Whittaker modules and obtain the necessary and sufficient conditions for the irreducibility of these modules. [ABSTRACT FROM AUTHOR]

Let A be a commutative noetherian non-positive DG-ring, ̄ an ideal of Ā : = H 0 (A) , and M ∈ D (A). In this paper, we introduce the notion of cohomological dimension of DG-modules and investigate the interplay between cd A ( ̄ , M) : = sup { cd Ā ( ̄ , H n (M)) + n | n ∈ ℤ } and sup R Γ ̄ (M). It is shown that cd A ( ̄ , M) = sup R Γ ̄ (M) for any 0 ≇ M ∈ D f − (A). As an application, we recover a DG-version of Grothendieck's vanishing and non-vanishing theorems for local cohomology. We also study the cohomological dimension of Koszul DG-modules and get some interesting results. [ABSTRACT FROM AUTHOR]

The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson's lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated. [ABSTRACT FROM AUTHOR]

The Drazin inverse is connected with the notion of index and core-nilpotent decomposition whenever it is discussed in the context of ring of matrices over complex field. In the absence of Drazin inverse for a given element from an arbitrary associative ring (not necessarily with unity), in this paper, the notion of right (left) core-nilpotent decomposition has been introduced and established its relations with right (left) π -regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) π -regular if and only if it has a right (left) core-nilpotent decomposition. In the process, several properties of sharp order in an associative ring are studied and with the help of the same, new characterizations of Drazin inverse over an associative ring are obtained and the relation between core-nilpotent decomposition and the Drazin inverse is obtained. [ABSTRACT FROM AUTHOR]

Let R be a ring with nonzero identity. The coidempotent graph of R , denoted it by G I d − (R) , has its set of vertices equal to the set of all elements of R ; distinct vertices x , y are adjacent in G I d − (R) if and only if x − y or y − x is an idempotent of R. In this paper, we study some basic properties of G I d − (R) and find some results about the ring-theoretic properties of R and graph-theoretic properties of G I d − (R). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the algebraic structure of 2 r 8 s -additive codes, where r and s are nonnegative integers, 2 (respectively, 8 ) denotes the finite field of order 2 (respectively, 8). We first give the generator polynomials of additive cyclic codes over 8 and then the generator polynomials of additive cyclic codes over 2 8 is also given. In addition, we introduce a linear map W : 2 8 → 2 , and study its properties. What's more, the dual of additive cyclic codes over 2 8 are investigated as well. And we get that the duals of any additive cyclic codes over 2 8 are also additive cyclic codes. Finally, separable 2 8 -additive cyclic codes are investigated. [ABSTRACT FROM AUTHOR]

Given a graph G whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling of G by the elements of R so that the difference of labels on adjacent vertices is an element of the corresponding edge ideal. The set of all generalized splines on a graph G with base ring R has a ring and an R -module structure. In this paper, we focus on the freeness of generalized spline modules over certain graphs with the base ring R = k [ x 1 , ... , x d ] where k is a field. We first show the freeness of generalized spline modules on graphs with no interior edges over k [ x , y ] such as cycles or a disjoint union of cycles with free edges. Later, we consider graphs that can be decomposed into disjoint cycles without changing the isomorphism class of the syzygy modules. Then we use this decomposition to show that generalized spline modules are free over k [ x , y ] and later we extend this result to the base ring R = k [ x 1 , ... , x d ] under some restrictions. [ABSTRACT FROM AUTHOR]

Recently, symplectic totally isotropic subspace inclusion graph ℐ n () on a 2 ν -dimensional symplectic space was introduced in [J. He, S. Zhang and G. Zhang, Symplectic totally isotropic subspace inclusion graph, Linear Multilinear Algebra (2022), doi:10.1080/03081087.2022.2112016], which is a graph whose vertices are all totally isotropic subspaces of and two distinct vertices are adjacent if and only if one is contained in the other. In that paper, the authors studied the diameter, girth, clique number and chromatic number of ℐ n () and also studied some other properties of ℐ n (). In this paper, we determine the automorphism group of ℐ n (). [ABSTRACT FROM AUTHOR]

This paper focuses on the study of ϕ-weak global dimensions in the context of polynomial rings and ϕ-Prüfer rings. We explore new properties of these dimensions and extend the Hilbert syzygy theorem to ϕ-weak global dimensions of rings. We also determine the ϕ-weak global dimension for certain types of ϕ-Prüfer rings. Key concepts such as ϕ-flat modules, ϕ-injective modules, and ϕ-torsion modules are discussed, along with their hereditary properties in PN-rings. This paper includes several theorems and lemmas that provide insights into the ϕ-weak global dimensions and their implications in the field of ring theory. [ABSTRACT FROM AUTHOR]

An integral domain R is said to be locally G-Dedekind if the Gorenstein global dimension of R is at most one for each maximal ideal . In this paper, we show that an integral domain R is not necessarily a G-Prüfer even if R is a locally G-Dedekind domain, which gives a negative answer to the question raised by the first author. It follows that the localization of the G-Prüfer domain differs from the classical case of the Prüfer domain. We also study coherent locally G-Dedekind domains, called almost G-Dedekind domains. The almost G-Dedekind domains need not be integrally closed and fill the gap between the G-Dedekind domains and the G-Prüfer domains. Various examples are provided to illustrate the new concept. [ABSTRACT FROM AUTHOR]

In this paper, we introduce multi-layer quivers and show how to construct an (n + 1) -slice algebra of infinite type from an n -slice algebra of infinite type using the bound quivers. This leads to constructing (n + 1) -slice algebras of infinite type as matrix algebra and as tensor algebra of an n -slice algebra and equivalences of their module categories as the module categories of diagram of some quiver of type A ̃ n + 1 . [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of ring called w p - S -Noetherian ring, which is a weak version of S -Noetherian ring property and study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamation of rings. Furthermore, we define the concept of nonnil w p - S -Noetherian ring property which is a generalization of the w p - S -Noetherian domain property and establish a characterization of this notion using pullbacks. [ABSTRACT FROM AUTHOR]

Co-Gorenstein algebras were introduced by Beligiannis in [A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and co-stabilization, Comm. Algebra28(10) (2000) 4547–4596]. In [S. Kvamme and R. Marczinzik, Co-Gorenstein algebras, Appl. Categorical Struct.27(3) (2019) 277–287], the authors propose the following conjecture (co-GC): if Ω n (m o d A) is extension closed for all n ≤ 1 , then A is right co-Gorenstein, and they prove that the generalized Nakayama conjecture implies the co-GC, also that the co-GC implies the Nakayama conjecture. In this paper, we characterize the subcategory Ω ∞ (m o d A) for algebras of Ω n -finite representation type. As a consequence, we characterize when a truncated path algebra is a co-Gorenstein algebra in terms of its associated quiver. We also study the behavior of Artin algebras of Ω ∞ -infinite representation type. Finally, an example of a non-Gorenstein algebra of Ω ∞ -infinite representation type and an example of a finite dimensional algebra with infinite ϕ -dimension are presented. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an affine cellular algebra A is semisimple if and only if the scheme associated to A is reduced and 0-dimensional, and the bilinear forms with respect to all layers of A are invertible. Moreover, if the ground ring is a perfect field, then A is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple. [ABSTRACT FROM AUTHOR]

Let h = ∏ i = 1 t p i s i be its decomposition into a product of powers of distinct primes, and ℤ h be the residue class ring modulo h. Let ℤ h n be the n -dimensional row vector space over ℤ h . A generalized Grassmann graph over ℤ h , denoted by G r (m , n , ℤ h) ( G r for short), has all m -subspaces of ℤ h n as its vertices, and two distinct vertices are adjacent if their intersection is of dimension > m − r , where 2 ≤ r ≤ m + 1 ≤ n. In this paper, we determine the clique number and geometric structures of maximum cliques of G r . As a result, we obtain the Erdős–Ko–Rado theorem for ℤ h n . [ABSTRACT FROM AUTHOR]

Let G be a finite group and Irr(G) be the set of irreducible characters of G. The number cod(χ) = | G : ker χ | / χ (1) is called the character codegree of χ ∈ Irr(G). In this paper, we show that if G has at most two composite character codegrees, then G is solvable. We also obtain sufficient numerical conditions on the character codegrees and character degrees for the solvability and supersolvability of a group. [ABSTRACT FROM AUTHOR]

In this paper, the author gives a complete set of simple Yetter–Drinfeld modules over the Suzuki algebra A N 2 n + 1 μ λ and investigates the Nichols algebras over those irreducible Yetter–Drinfeld modules. The finite-dimensional Nichols algebras of diagonal type are of Cartan type A 1 , A 1 × A 1 , A 2 , Super type A 2 (q ; 2) and the Nichols algebra (8). And the involved finite-dimensional Nichols algebras of non-diagonal type are 1 2 , 4 m and m 2 -dimensional. The left three unsolved cases are set as open problems. In particular, all finite-dimensional Nichols algebras are given for simple Yetter–Dinfeld modules over  A 1 3 + λ . [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to be C S S -subgroup of G if there exists a normal subgroup K of G such that G = H K and H ∩ K is S S -quasinormal in G. In this paper, we investigate the p -nilpotency of the finite groups under the assumption that there is a subgroup D of P such that 1 < | D | < | P | and every subgroup of P with order | D | or 2 | D | (whenever | D | = 2) is C S S -subgroup, where p is a prime dividing the order of G and P is a Sylow p -subgroup of G. As an application of the results, a related problem posed by Heliel et al. in [Finite groups with minimal CSS-subgroups, Eur. J. Pure Appl. Math.14 (2021) 1002–1014] is solved. [ABSTRACT FROM AUTHOR]

A multiplication on an Abelian group G is a homomorphism μ : G ⊗ G → G. The set Mult G of all multiplications on an Abelian group G itself is an Abelian group with respect to addition; this group is called the multiplication group of G. In the paper, the class 1 of quotient divisible Abelian groups of rank 1 is considered. The group Mult G is described for G ∈ 1. In the group Mult G , multiplications defining comparable rings on G are described for any group G ∈ 1. The isomorphism problem is solved in 1 : multiplications defining isomorphic rings on G are described for any G ∈ 1. [ABSTRACT FROM AUTHOR]

Let E be a vector bundle of rank r on a smooth complex projective variety X. In this paper, we compute the nef and pseudoeffective cones of divisors on the Grassmann bundle Gr X (k , E) parametrizing k -dimensional subspaces of the fibers of E , where 1 ≤ k ≤ rk(E), under assumptions on X as well as on the vector bundle E. In particular, we show that nef cone and the pseudoeffective cone of Gr X (k , E) coincide if and only if nef cone and pseudoeffective cone of X coincide under the assumption that E is a slope semistable bundle on X with c 2 (End(E))=0. We also discuss about the nefness and ampleness of the universal quotient bundle Q k on Gr X (k , E). [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S -primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S -primary ideal if there exists s ∈ S such that whenever 0 ≠ a b ∈ I for a , b ∈ R , then s a ∈ I or s b ∈ I. The relationships among S -prime, S -primary, weakly S -primary and S - n -ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S r -primary ideals are characterized where S r = { 1 , r , r 2 , ... }. Several properties, characterizations and examples concerning weakly S -primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S -decomposable ideals and S -weakly Laskerian rings which are generalizations of S -decomposable ideals and S -Laskerian rings are introduced. [ABSTRACT FROM AUTHOR]

In this paper, we provide the necessary and sufficient conditions for the edge binomials of the tree forming a d -sequence in terms of the degree sequence notion of a graph. We study the regularity of powers of the binomial edge ideals of trees generated by d -sequence edge binomials. [ABSTRACT FROM AUTHOR]

For a rooted cluster algebra (Q) over a valued quiver Q , a symmetric cluster variable is any cluster variable belonging to a cluster associated with a quiver σ (Q) , for some permutation σ. The subalgebra of (Q) generated by all symmetric cluster variables, is called the symmetric mutation subalgebra and is denoted by ℬ (Q). In this paper, we identify the class of cluster algebras that satisfy ℬ (Q) = (Q) , which contains almost every quiver of finite mutation type. In the process of proving the main result, we provide a classification of quivers mutations classes that relates their maximum weights to the shapes of the initial quivers. Furthermore, some properties of symmetric mutation subalgebras are given. [ABSTRACT FROM AUTHOR]

Let X be a nonempty set and T (X) denote the full transformation semigroup on X. For a fixed nonempty subset Y of X , let PG Y (X) = { α ∈ T (X) : α | Y ∈ G (Y) } , where G (Y) is a permutation group on Y. Then PG Y (X) is a regular submonoid of T (X). In this paper, we describe all intra-regular and unit regular elements of PG Y (X) and give necessary and sufficient conditions for PG Y (X) to be intra-regular and unit regular. We also count the number of these elements when X is a finite set. Moreover, we classify the maximal subsemigroups of PG Y (X) and prove that these maximal subsemigroup coincide with the maximal regular subsemigroups of PG Y (X) when X / Y is a finite set. [ABSTRACT FROM AUTHOR]

In this paper, we characterize and classify the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of SL 3 (k) and SL 4 (k) where k is a finite field of odd characteristic. We provide some generalized results for SL n (k). [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with a nonzero identity and M an R -module. Set ϕ -tor  (M) = { x ∈ M | s x = 0  for some  s ∈ R \ Nil (R) } , if ϕ - tor (M) = M then M is called a ϕ -torsion module. An R -module M is said to be ϕ -flat, if 0 → A ⊗ R M → B ⊗ R M → C ⊗ R M → 0 is an exact R -sequence, for any exact sequence of R -modules 0 → A → B → C → 0 , where C is ϕ -torsion. In this paper, we study some new properties of ϕ -flat modules. Then we introduce and study the class of ϕ - P -flat modules which is a generalization of ϕ -flat modules and P -flat modules. Finally, we give some new characterizations of the ϕ -von Neumann regular ring and its transfer to various contexts of constructions such as the amalgamation of rings along an ideal and trivial ring extension. [ABSTRACT FROM AUTHOR]

In this paper, some properties of vertex algebras acted by a Hopf algebra will be given. We prove that V # H is an -local vertex algebra, where V is an H -module vertex algebra. In addition, we extend the -locality to the quantum vertex algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study gyro-groups associated to groups, group extensions admitting gyro-sections, and corresponding co-homologies. We also describe the obstructions in terms of co-homomology. The notion of gyro-Schur multiplier and that of gyro-Milnor K 2 group are introduced. [ABSTRACT FROM AUTHOR]

A proper ideal I of a commutative ring R is said to be a strongly quasi-primary ideal if, whenever a , b ∈ R with a b ∈ I , then a 2 ∈ I or b ∈ I (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of R , called the class of quasi- n -ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of R is a maximal or a minimal ideal of R. Many examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of G is said to be a BNA-subgroup of G if either H x = H or x ∈ 〈 H , H x 〉 for all x ∈ G. A subgroup H of G is said to be a weakly BNA-subgroup of G if there exists a normal subgroup T of G such that G = H T and H ∩ T is a BNA-subgroup of G. In this paper, we investigate the structure of a finite group G under the assumption that every minimal subgroup of G not having a supersolvable supplement in G is a weakly BNA-subgroup of G. [ABSTRACT FROM AUTHOR]

For a connected graph G (V , E) a clique S ⊆ V (G) is twin-free if every pair of elements of S have distinct closed neighborhoods and the number of elements in a twin-free clique of maximum cardinality is called twin-free clique number of G. The annihilator graph A G (R) of a commutative and unital ring R is a graph whose vertices are all non-zero zero-divisors of R and there is an edge between two distinct vertices a , b if and only if ann (a) ∪ ann (b) is properly contained in ann (a b). In this paper, twin-free clique number of A G (R) is computed and as an application the strong metric dimension of A G (R) is characterized. Among other things, for a reduced ring R , the forcing strong metric dimension of A G (R) is given. [ABSTRACT FROM AUTHOR]

Let R be a ring of dimension d ≥ 1 containing ℚ and A = R [ X , Y , Z , W ] (X Y − Z W) . This paper examines the question of efficient generation of ideals in A. [ABSTRACT FROM AUTHOR]

Let and ′ be two ∗-Jordan algebras with identities I and I ′ , respectively, and e a nontrivial ∗-idempotent in . In this paper, we study the characterization of multiplicative ∗-Jordan-type maps. In particular, we provide a characterization in the case of unital prime associative algebra endowed with an involution. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of cotorsion pairs in m -cluster categories of type A , denoted by A n − 1 m , via certain configurations of m -diagonals, called Ptolemy diagrams. As applications, we classify m -rigid subcategories of A n − 1 m , which gives Jacquet–Malo's classification of m -cluster tilting subcategories of A n − 1 m . When m = 1 , this generalizes the work of Holm et al. [Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A n , J. Algebraic Combin. 34 (3) (2011) 507–523] for the classification of torsion pairs in cluster categories of type A n − 1 . [ABSTRACT FROM AUTHOR]