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AbstractLet H and K be groups. In this paper we introduce a concept of determinant for endomorphisms of H × K and some concepts of incompatibility for group pairs as a measure of how far H and K are from being isomorphic. With the aid of the tools developed, we give a characterisation of invertible automorphisms of H × K by means of their determinants and an explicit description of Aut(H×K) as a group of 2-by-2 matrices, in case H or K belong to some relevant classes of groups. Many theoretical and practical applications of the determinants will be presented, together with examples and an analysis on some computational advantages of the determinants. In particular, we give a matrix characterisation of the whole automorphism group of some relevant classes of groups. [ABSTRACT FROM AUTHOR]

AbstractThis paper introduces and studies the weak π-Rickart ring property, which represents a new extension of the Rickart ring condition. Regularity, extending and prime-like conditions for the weak π-Rickart rings are defined and shows the implications between these conditions and their links with Rickart, p.q.-Baer, regular and biregular conditions. We include applications to polynomial and formal power series extensions and to triangular matrix rings, with examples given to demonstrate the results. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we compare the regularities of symbolic and ordinary powers of edge ideals of weighted oriented graphs. For any weighted oriented complete graph Kn, we show that reg(I(Kn)(k))≤reg(I(Kn)k) for all k≥1. Also, we give explicit formulas for reg(I(Kn)(k)) and reg(I(Kn)k), for any k≥1. As a consequence, we show that reg(I(Kn)(k)) is eventually a linear function of k. For any weighted oriented graph D, if V+ are sink vertices, then we show that reg(I(D)(k))≤reg(I(D)k) with k = 2, 3 and equality cases studied. Furthermore, we give formula for reg(I(D)2) in terms of reg(I(D)(2)) and regularity of certain induced subgraphs of D. Finally, we compare the regularity of symbolic powers of weighted oriented graphs D and D′, where D′ is obtained from D by adding a pendant. [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite π -solvable group. In this paper, we show that all monomial irreducible Isaacs π -partial character degrees of G are π -number, if and only if G has a normal Hall π′ -subgroup. Furthermore, let G be a finite π -separable group and let p be prime in π . We prove that if p divides φ(1) for all non-linear φ∈Iπ(G) , then G has a normal p-complement. [ABSTRACT FROM AUTHOR]

AbstractLet POPDIn (PODIn) be the semigroup consisting of all injective orientation-preserving (order-preserving) and order-decreasing transformations on the finite chain {1<⋯

AbstractLet p1, p2 be two distinct prime integers, let n be a positive integer, n≥3 and let ξn be a primitive root of order n of the unity. In the 3rd section of this paper we obtain a complete characterization for a quaternion algebra H(p1,p2) to be a division algebra over the nth cyclotomic field Q(ξn), when n∈{3,4,6,7,8,9,11,12} and we also obtain a characterization for a quaternion algebra H(p1,p2) to be a division algebra over the nth cyclotomic field Q(ξn), when n∈{5,10}. In the 4th section we obtain a complete characterization for a quaternion algebra HQ(ξn)(p1,p2) to be a division algebra, when n=lk, with l a prime integer, l≡3 (mod 4) and k a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra HQ(ξl)(p1,p2) to be a division algebra, when l is a Fermat prime number. [ABSTRACT FROM AUTHOR]

AbstractRings all of whose right ideals are automorphism-invariant are called right a-rings. It is shown that (1) local right a-rings are left duo, (2) local rings whose left ideals of R are invariant under all units of R are left duo, (3) a right a-ring R with dim(RR)=1 is duo. In this paper, the question of when is a left a-ring a right a-ring is also addressed. It is shown that (4) if R is a quasi-Frobenius ring or right (or left) noetherian/artinian ring, then R is a right a-ring if and only if R is a left a-ring. It is also proved that (5) R is a right a-ring with ACC on right annihilators iff R is a left a-ring with ACC on left annihilators, questions (partially) answering the different question on this topic. [ABSTRACT FROM AUTHOR]

AbstractA ring R with center C is said to be centrally essential if the module RC is an essential extension of the module CC . In this paper, we study properties of ideals of centrally essential rings, centrally essential quaternion algebras, and group rings of Hamiltonian groups. [ABSTRACT FROM AUTHOR]

Let D be an integral domain and D [ ​ [ x ] ​ ] be the power series ring over D. In this paper, we study the primality of x − α in D [ ​ [ x ] ​ ] , where α ∈ D . For this purpose, we generalize the definition of a prime element as follows. For a positive integer N, a nonzero nonunit α ∈ D is called an N-prime element if for any a , b ∈ D , α N | ab implies α | a or α | b . We prove that if α is an N-prime element for some N, then x − α is a prime element in D [ ​ [ x ] ​ ] . Surprisingly, it is shown that the converse also holds when D is a PID, a valuation domain, or a Dedekind domain. In other words, when D is a PID, a valuation domain, or a Dedekind domain, a necessary and sufficient condition for x − α to be a prime element in D [ ​ [ x ] ​ ] is α is an N-prime element in D for some N. This however does not hold for other types of integral domains such as UFDs or Krull domains. We also investigate the N-prime property in an arbitrary integral domain and give other (equivalent) conditions for an element α in the aforementioned types of integral domains to be an N-prime element. [ABSTRACT FROM AUTHOR]

Let (A , B , C) be a recollement of extriangulated categories. In this paper, we provide bounds on the coresolution dimensions of the subcategories involved in A , B , and C . We show that a hereditary cotorsion pair in B can induce hereditary cotorsion pairs in A and C under certain conditions. Besides, we construct recollements of abelian categories from a recollement of extriangulated categories with respect to the hearts of cotorsion pairs. [ABSTRACT FROM AUTHOR]

For a character χ of a finite group G, the number cod (χ) = [ G : Ker χ ] χ (1) is called the codegree of χ. In this paper, we show that if for every non-principal irreducible characters χ and ϕ of G with cod (χ) ≠ cod (ϕ) , the greatest common divisor of cod (χ) and cod (ϕ) is a prime-power, then either G is solvable or G / Sol (G) is isomorphic to one of the groups Alt 5 or PSL 2 (8) , where Sol (G) is the solvable radical of the group G. [ABSTRACT FROM AUTHOR]

It is known that the degree of irrationality irr (X) of a smooth surface X of degree d ≥ 5 in P 3 is at least d – 2. In this paper, we study a Galois rational map f : X − − → P 2 of degree d – 2, and we show that the Galois group of f is a cyclic group, and f is defined by the family of lines intersecting two fixed skew lines. In addition, we give a necessary and sufficient condition that X has skew lines which give a Galois rational map of degree d – 2 by using automorphisms of X. [ABSTRACT FROM AUTHOR]

In this paper we study subinjectivity domains of various R-modules and inclusion relations between these domains. We show that if the class of all subinjectivity domains is linearly ordered, then R is right Noetherian, and is either a right V-ring or a ring with unique noninjective simple module U. For the latter case, if U is projective but not indigent, then there exists a ring decomposition R = S × T such that S is a semisimple Artinian ring and T is an indecomposable right Artinian right hereditary ring. Also, in that case, if the subinjectivity domain of U is the only middle subinjectivity domain, then R is right Artinian. [ABSTRACT FROM AUTHOR]

The conjugacy relation is an important tool in the study of group theory and now has been generalized to semigroups in several methods. In particular, the p-conjugacy, o-conjugacy, c-conjugacy, i-conjugacy, n-conjugacy and r-conjugacy in semigroups are introduced and investigated extensively and some problems related to these conjugacy relations are proposed. The aim of this paper is to continue the study of conjugacy relations in semigroups around some of these problems. We first point out that p-conjugacy in a semigroup that is embeddable in a group, is not necessarily transitive. Then we obtain a simple sufficient condition under which the o-conjugacy is the universal relation and prove that this condition is also necessary in some natural classes of semigroups. Finally, we explore some properties of r-conjugacy in restriction semigroups, which extends some results on i-conjugacy in inverse semigroups. Our results have answered or partially answered some problems raised by Araújo, Kinyon, Konieczny and Malheiro in [Proc.Roy.Soc.Edinburgh Sect.A 147, 1169–1214, 2017] and [J.Algebra 533, 142–173, 2019]. [ABSTRACT FROM AUTHOR]

Let M be a right R-module. We call a summand A of M a special summand if there exists a summand B of M with A ≅ B and A ∩ B = 0 . An internal direct sum ⊕ i ∈ I A i of submodules Ai of M is called a special local summand if each Ai is a special summand and ⊕ i ∈ F A i is a summand of M for any finite subset F ⊆ I . In this paper we give two new classes of modules for which the finite exchange property implies the full exchange property, namely, modules whose special local summands are summands, and C4-modules for which the union of any chain of special summands is a summand. More precisely, we show that if either M is a module whose special local summands are summands or M is a C4-module such that the union of any chain of special summands of itself is a summand, then M has the finite exchange property if and only if it has the full exchange property if and only if it is a clean module. As immediate applications of the aforementioned results, we generalize and extend many of the fundamental results on the subject of exchange modules. [ABSTRACT FROM AUTHOR]

Pre-Novikov algebras and quasi-Frobenius Novikov algebras naturally appear in the theory of Novikov bialgebras. In this paper, we show that there is a natural pre-Novikov algebra structure associated to a quasi-Frobenius Novikov algebra. Then we introduce the definition of double constructions of quasi-Frobenius Novikov algebras associated to two pre-Novikov algebras and show that it is characterized by a pre-Novikov bialgebra. We also introduce the notion of pre-Novikov Yang-Baxter equation, whose symmetric solutions can produce pre-Novikov bialgebras. Moreover, the operator forms of pre-Novikov Yang-Baxter equation are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we characterize a relative Rota-Baxter operator on an n-Lie superalgebra with a representation (called an n-LSRep pair) as a Maurer-Cartan element of a certain Lie n-algebra. Then, we introduce the notions of n-pre-Lie superalgebras and symplectic n-Lie superalgebras and give some related results based on relative Rota-Baxter operators. In the next part, we give the cohomologies of relative Rota-Baxter operators on n-LSRep pairs and study infinitesimal deformations and extensions of finite order deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Finally, we build the relationship between the cohomology groups of relative Rota-Baxter operators on n-LSRep pairs and those on (n + 1) -LSRep pairs by certain linear functions. [ABSTRACT FROM AUTHOR]

The notion of matching Rota-Baxter algebras was recently introduced by Gao, Guo, and Zhang motivated by the study of algebraic renormalization of regularity structures. In this paper, we introduce matching relative Rota-Baxter algebras that are closely related to matching dendriform algebras. We define the cohomology of a matching relative Rota-Baxter algebra using the classical Hochschild cohomology and a new cohomology induced by the matching operators. As an application, we show that our cohomology governs the formal deformation theory of the matching relative Rota-Baxter algebra. Finally, we define the cohomology of a matching dendriform algebra and study homotopy matching dendriform algebras. [ABSTRACT FROM AUTHOR]

Let G be a group and A, B, C be subgroups of G. In this paper, we prove that if G = AB ∪ BC , then G = AB or G = BC. But, in general, G = AB ∪ AC does not imply that G = AB or G = AC. We also prove that if G is a 2-decomposed group, and G is a Sylow tower group or G is nilpotent-by-nilpotent, then G = AB ∪ AC implies that G = AB or G = AC. Other interesting special cases are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, the concept of iso-DICC modules is introduced and extensively investigated. These modules are the generalization of DICC, iso-Noetherian and iso-Artinian modules. We prove that any direct sum of a projective iso-simple and an iso-DICC module is iso-DICC. Unlike DICC rings, we show that iso-DICC rings don't have finite Goldie dimension, but if R is a semiprime right iso-DICC ring and I is an annihilator ideal of R, then either I or R I has finite Goldie dimension. Furthermore, we investigate the Krull dimension of iso-Noetherian modules and we give an answer to an open problem on this topic. Finally, we generalize some results on semiprime right iso-Artinian rings to a certain class of iso-Artinian modules. We provide sufficient conditions for iso-Artinian modules to have finite Goldie dimension. Also, we show that any semiprime FQS iso-Artinian module, is non-M-singular. [ABSTRACT FROM AUTHOR]

Let Zp be a group of order p and G ≀ Z p the regular wreath product of the group G with Zp. A Fitting class F is said to be X -quasinormal if there exists a natural number m such that G m ≀ Z p ∈ F whenever F ⊆ X , p is a prime, G ∈ F and G ≀ Z p ∈ X . In this paper, we generalize the well known theorem of Blessenohl and Gaschütz and prove that the intersection of any set of nonidentity X -quasinormal Fitting classes is a nonidentity X -quasinormal Fitting class. For a Fitting class F , we also discuss the relation between its X -normality and its X -quasinormality, and give the condition of X * = X ⊛ , where " * " is Lockett star operation and X ⊛ is the smallest nonidentity X -quasinormal Fitting class. [ABSTRACT FROM AUTHOR]

A graph Γ is said to be 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs of Γ . In this paper, we give a group-theoretic characterization of those connected 2-arc-transitive graphs which admit a vertex-transitive simple group. [ABSTRACT FROM AUTHOR]

AbstractUsing the natural Z-grading, Graded Simplicity and Graded Uniqueness Theorems have been obtained in the past for Leavitt path algebras over fields and commutative unital rings. In this paper we extend some of these results to the more general setting of Leavitt path algebras over Clifford semifield and k-semifield. Clifford semifield and k-semifield are natural generalizations of bounded distributive lattices as well as of fields. As evident from some previous works, Leavitt path algebra in these settings is not exactly the same as that in case of fields and commutative rings. Here we characterize graded full k-ideal simplicity of Leavitt path algebra over Clifford semifield and graded k-ideal simplicity over k-semifield by the graph theoretic properties of the underlying directed graph. We apply our simplicity theorems on the Leavitt path algebra of rose graph with one petal to establish that S[x,x−1] is graded full k-ideal simple (respectively, graded k-ideal simple) for any Clifford semifield (respectively, k-semifield) S. Finally we obtain versions of Graded Uniqueness Theorem for Leavitt path algebras with coefficients in Clifford semifield and k-semifield. [ABSTRACT FROM AUTHOR]

A ring idempotent e ∈ R is said to be quarter-central (or q-central for short) if e R (1 − e) R e = 0 . If all idempotents in a ring R are q-central, R is said to be a quarter-abelian ring (or more simply, a q-abelian ring). In this paper, we show that such a ring R is characterized by the property that [ e , R ] [ e , R ] = 0 for all idempotents e ∈ R , where [ x , y ] denotes the additive commutator x y − y x . For any nonzero ring S and any integer n ≥ 3 , we show that the ring T n (S) of n × n upper triangular matrices over S is not q-abelian. On the other hand, T 2 (S) is q-abelian iff S is abelian (in the classical sense that all idempotents are central in S ). A final section of this paper relates q-central idempotents to the notion of exchange rings and the study of regular, unit-regular, and strongly regular elements in arbitrary rings. From the viewpoint of currently prevailing generalizations of abelian rings in the literature, q-abelian rings are situated between the class of semiabelian rings and the class of strongly IC rings. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function m (G) = ∑ g ∈ G 1 / o (g) , where o(g) denotes the order of g ∈ G . In this paper, we prove that if m (G) < m (A 5) , m (G) < m (A 4) , m (G) < m (S 3) , m (G) < m (Q 8) or m (G) < m (C 2 × C 2) , then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively. [ABSTRACT FROM AUTHOR]

This work is devoted to the complete classification of solvable Leibniz algebras with the naturally graded quasi-filiform non-Lie nilradical. It should be noted that the class of naturally graded quasi-filiform Leibniz algebras is sufficiently large, and solvable Leibniz algebras with the nilradical which is isomorphic to several of these algebras are classified in a series of works. This paper considered all omitted cases and completed the problem of classifying solvable Leibniz algebras with naturally graded quasi-filiform non-Lie nilradical. At the end of the paper we combine all results obtained to date, and provide a complete list of solvable Leibniz algebras, whose nilradical is a naturally graded quasi-filiform Leibniz algebra. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra s c h n : = s l 2 ⋉ h n is the semi-direct product of the Lie algebra s l 2 with the n-th Heisenberg Lie algebra h n . In this paper, we classify the s c h n -modules (n ≥ 2) that are free of rank 1 when restricted to the Cartan subalgebra (modulo center). Furthermore, the simplicities of these modules are determined. [ABSTRACT FROM AUTHOR]

Let G be a finite group and χ ∈ Irr (G) . The codegree of χ is defined as cod (χ) = | G : ker (χ) | / χ (1) and cod (G) = { cod (χ) | χ ∈ Irr (G) } is called the set of codegrees of G. In this paper, we show that the set of codegrees of U 4 (2) , S 4 (q) (q = 2 f ≥ 4) , U 4 (3) , 2 F 4 (2) ′ , J 3 , G 2 (3) , A 9 , J 2 , PSL (4 , 3) , McL , S 4 (5) , G 2 (4) , HS , O ′ N and M 24 determines the group up to isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of monomials in the q-characters of the fundamental modules over a quantum affine algebra of untwisted type C. As a result, we give simple closed formulae for the q-characters of the fundamental modules in terms of sequences of vertices in R 2 , so-called paths, with an admissible condition. This may be viewed as a type C analog of the path description of q-characters in types A and B due to Mukhin–Young. [ABSTRACT FROM AUTHOR]

In this paper, we classify eight-dimensional non-solvable Lie algebras that support a symplectic structure. As well as a complete classification is given, up to symplectomorphism, of eight-dimensional symplectic non-solvable Lie algebras. [ABSTRACT FROM AUTHOR]

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A (Δ) = k [ x 1 , ... , x n ] / 〈 x 1 2 , ... , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A (Δ) in terms of the simplicial complex Δ. We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A (Δ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization. [ABSTRACT FROM AUTHOR]

Let G be an arbitrary group. For any subsets A and B of G, let A * B = { a * b ; a ∈ A , b ∈ B } where ' * ' is the binary operation on G. By μ G (r , s) , we denote the minimum cardinality of the set A * B , where | A | = r and | B | = s . In 2003, Eliahou et al. proposed a conjecture that for every finite group G of order g and every pair of integers r , s with 1 ≤ r , s ≤ g , k G (r , s) ≤ μ G (r , s) , where k G (r , s) = min d ∈ H (G) { (⌈ r d ⌉ + ⌈ s d ⌉ − 1) d } and H (G) is the set of orders of finite subgroups of G. In 2003, Eliahou et al. verified the conjecture for dihedral group D q of index q = p v . In this paper, we determine the upper bound for the size of μ Q 4 n (r , s) and μ U m , n (r , s) , where Q 4 n = 〈 a , b : a 2 n = e , b 2 = a n , a b = b a − 1 〉 is a dicyclic group of order 4n and U m , n = 〈 a , b : a 2 n = e , b m = e , a b a − 1 = b − 1 〉 and verify the conjecture proposed by Eliahou et al. [7] for Q 4 n for n to be a prime power. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the notion of relative Rota-Baxter operators on Hom-Lie triple systems and define a cohomology of relative Rota-Baxter operators on Hom-Lie triple systems with coefficients in a suitable representation. Furthermore, we introduce and study Hom-pre-Lie triple systems as the underlying structure of relative Rota-Baxter operators on Hom-Lie triple systems. Finally, we provide relationships between relative Rota-Baxter operators on Hom-Lie algebras and associated Hom-Lie triple systems. [ABSTRACT FROM AUTHOR]

Let (hyp o n , ♯) be the hypoplactic monoid of finite rank n with Schützenberger's involution ♯ . In this paper, we exhibit a faithful representation of (hyp o n , ♯) as an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. We then give a transparent combinatorial characterization of the word identities satisfied by (hyp o n , ♯) . Further, we prove that (hyp o n , ♯) is non-finitely based if and only if n = 2, 3 and give a polynomial time algorithm to check whether a given word identity holds in (hyp o n , ♯) . Communicated by Scott Chapman [ABSTRACT FROM AUTHOR]

In the paper, we study the isotropy groups of K [ x ] if D is a simple derivation. We first study the derivation of the form: D = ∂ x 1 + ∑ i = 2 n (a i x i + b i) ∂ x i with a i , b i ∈ K [ x 1 , ... , x i − 1 ] for all 2 ≤ i ≤ n . We prove that Aut (K [ x ]) D = { i d } if D is simple with deg x i − 1 a i ≥ 1 for all 2 ≤ i ≤ n or a i ∈ K * for all 3 ≤ i ≤ n or 0 ≤ deg x j b i ≤ deg x j a i for all 2 ≤ j ≤ i − 1 , 3 ≤ i ≤ n . Thus, we conjecture that Aut (K [ x ]) D = { i d } if D is simple with ∏ i = 2 n a i ≠ 0 , a i , b i ∈ K [ x 1 , ... , x i − 1 ] for all 2 ≤ i ≤ n . Then we prove that Aut (K [ x ]) D = { (x 1 , ... , x n − 1 , x n + c) | c ∈ K } if D = (x 1 s x 2 + g) ∂ x 1 + x 1 s − 1 ∂ x 2 + g 3 ∂ x 3 + ⋯ + g n ∂ x n or D = (x 1 s x 2 t + c) ∂ x 1 + x 1 r ∂ x 2 + g 3 ∂ x 3 + ⋯ + g n ∂ x n with g ∈ K [ x 1 ] , deg (g) ≤ s , g (0) ≠ 0 and g i ∈ K [ x i − 1 ] \ K for all 3 ≤ i ≤ n . [ABSTRACT FROM AUTHOR]

Let G be a finite group and δ (G) = 1 | G | ∑ H ≤ G { | H | | H is non-cyclic}. In this paper, we show that some arithmetical conditions of δ (G) influence the structure of G. Firstly, we prove that if δ (G) < 13 3 , then G is solvable. Secondly, we determine the structure of finite groups with δ (G) ≤ 2 . Moreover, we prove that if δ (G) < 1 + 4 | G | , then G is supersolvable, and we also determine the structure of finite groups G with δ (G) = 1 + 4 | G | . Finally, we show that δ (G) < c does not imply the supersolvability of G for any constant c ∈ (1 , ∞) . [ABSTRACT FROM AUTHOR]

The present paper is devoted to studying 2-local superderivations on the twisted N = 2 superconformal algebra, the Ramond N = 2 super-Virasoro algebra and the super W -algebra S W (3 2 , 3 2) . It is proved that all 2-local superderivations on these Lie superalgebras are superderivations. [ABSTRACT FROM AUTHOR]

In this paper, we contribute to the theory of reduced G-perfection and horizontal linkage of modules over a commutative, Noetherian (typically, local) ring R, in the general setting where properties and operations are considered relatively to a semidualizing module C. We investigate when reduced GC-perfection is preserved by relative Auslander transpose, and how to numerically characterize horizontally linked modules. Moreover, we show how to produce reduced G C -perfect modules that are also C-k-torsionless ( k ≥ 0 is an integer) but fail to be G C -perfect, and we illustrate that, unlike the usual grade, the relative reduced grade depends on the choice of C. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the classifying functors W ¯ and Diag ° N are simplicially homotopy equivalent. Using this equivalency, we define classifying simplicial modules and give an alternative definition of (co)homology of crossed modules of algebras. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with unity. A unital left R-module M is said to be distributive if for every submodules S, T and U of M, the equality S ∩ (T + U) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for a direct sum of left R-modules to be distributive. This condition is given by the notion of splitting of submodules of the direct sum and the proof uses the notion of orthogonality, where both notions are discussed and revisited. [ABSTRACT FROM AUTHOR]

We study the Fano variety of k-planes Ω X (k) of a complete intersection X and show the surjectivity of the cylinder homomorphism in this setting, proving along the way the (classical) Hodge conjecture in cases where a numerical condition is satisfied. In particular, a new case is discovered in dimension 6. This work is a generalization of the second author's paper Motives of some Fano varieties. [ABSTRACT FROM AUTHOR]

The Frobenius-Perron dimension of a matrix, also known as the spectral radius, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, and find a way to calculate the Frobenius-Perron dimensions of these algebras satisfying the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximal number of loops at each vertex. Finally, we point out that there also exist infinite dimensional algebras whose Frobenius-Perron dimensions is equal to the maximal number of loops by giving an example. [ABSTRACT FROM AUTHOR]

In this paper, first we give the notion of a crossed homomorphism on a Lie triple system with respect to an action on another Lie triple system. We also construct an L ∞ -algebra using the higher derived brackets, whose Maurer-Cartan elements are crossed homomorphisms on Lie triple systems. Consequently, we obtain the twisted L ∞ -algebra that controls deformations of a given crossed homomorphism on Lie triple systems. Next we construct a cohomology theory for a crossed homomorphism on Lie triple systems and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Moreover we consider the relationship between cohomology groups of crossed homomorphisms on Lie algebras and associated Lie triple systems. [ABSTRACT FROM AUTHOR]

In this paper we consider the Br = Br ′ question for classifying stacks by various group schemes. These are algebraic stacks that do not necessarily admit a finite flat cover by a scheme for which Br = Br ′ holds, hence are not amenable to the usual argument of pushing forward a twisted vector bundle. We provide two classes of examples satisfying Br ≠ Br ′ that do not arise from the scheme case. Communicated by Tony Varilly-Alvarado [ABSTRACT FROM AUTHOR]

Consider a prime ring R which is noncommutative with the maximal left ring of quotients of R denoted by Q m l (R) and the extended centroid C. An additive map G : R → Q m l (R) satisfying G (g 2) + η g G (g) ∈ C for all g ∈ R , where η ∈ C is called a weak Jordan right η-centralizer. In this paper, we establish the characterization of weak Jordan right η-centralizers in prime rings. As an application, we describe X-generalized skew derivations which behave like weak Jordan right η-centralizers. [ABSTRACT FROM AUTHOR]

It is classical that certain truncations of the translation quiver Z Q appear in the Auslander-Reiten quiver. The stable n-translation quiver Z | n − 1 Q is introduced as a generalization of Z Q in studying higher representation theory. In this paper, we find conditions for a Hom-finite k-category to be realized as the bound path category of a convex full subquiver of a stable n-translation quiver. We show that the n-preinjective component and n-preprojective component of n-slice algebra are realized as the bound path categories of some truncations of Z | n − 1 Q o p . We also use Z | n − 1 Q o p to describe the νn-closure of Γ in the derived category. [ABSTRACT FROM AUTHOR]

Projectively coresolved Gorenstein flat modules were introduced recently by Saroch and Stovicek and were shown to be Gorenstein projective. While the relation between Gorenstein projective and Gorenstein flat modules is not well understood, the class of projectively coresolved Gorenstein flat modules is contained in the class of Gorenstein flat modules. This paper proves necessary and sufficient conditions for a module of finite Gorenstein flat dimension to be projectively coresolved Gorenstein flat, or of finite flat dimension. Stability results for the class of projectively coresolved Gorenstein flat modules are also established. [ABSTRACT FROM AUTHOR]

Green biset functors are introduced by Serge Bouc as a general framework to work with biset functors having an additional ring structure. Well-known examples include the functors of (fibered) Burnside rings, trivial source rings and character rings. Serge Bouc also introduced two generalizations of Burnside rings, the so-called slice and section Burnside rings which are also Green biset functors. In this paper, we classify simple modules over the Green biset functor of section Burnside rings. [ABSTRACT FROM AUTHOR]

The goal of this paper is to construct a semifree resolution for a non-negatively graded strongly commutative DG algebra B over the enveloping DG algebra B ⊗ A B , where A ⊆ B is a DG subalgebra and B is semifree over A. Our construction of such a semifree resolution, that we denote by (B , D) , uses the notions of reduced bar resolution and tensor algebra of the shift of the diagonal ideal. As an application of (B , D) , we provide a new characterization of naïve liftability of semifree DG B-modules. [ABSTRACT FROM AUTHOR]

Let G be a finite group, p a prime dividing | G | . In this paper, we study the structure of G, which satisfies the condition that any two irreducible characters in its principal p-block have distinct degrees. We provide a reduction of this question to simple groups and determine the structure of G for p = 2. Furthermore, we describe the structure of G in which any two characters of the principal p-block of G have coprime degrees. [ABSTRACT FROM AUTHOR]