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Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

In this paper, we shall study three types of Galois extensions B which are Azumaya algebras over its center C. (1) 1 , the class of Azumaya–Galois extensions; (2) 2 , the class of Galois extensions B of B G with Galois group G such that B G is a separable C G -algebra; and (3) 3 , the class of Galois extensions which are Azumaya algebras over its center C. Clearly, 1 ⊆ 2 ⊆ 3 . Examples are given to show the proper inclusion relationship 1 ⊂ 2 ⊂ 3 , and equivalent conditions are given under which a B ∈ 3 is in 1 and a B ∈ 2 is in 1 . [ABSTRACT FROM AUTHOR]

This paper details the diassociative analogue of results concerning the Schur multiplier and other extension-theoretic concepts that originate in group theory. We first prove that covers of diassociative algebras are unique. Second, we show that the multiplier of a diassociative algebra is characterized by the second cohomology group with coefficients in the field. Third, we establish criteria for when the center of a cover maps onto the center of the algebra. Along the way, we obtain a collection of exact sequences, characterizations, and a brief theory of unicentral diassociative algebras and stem extensions. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras. [ABSTRACT FROM AUTHOR]

Let H ≤ K be subgroups of a finite group G , then H is called strongly closed in K with respect to G if H g ∩ K ≤ H for every g ∈ G , and in particular, H is simply called strongly closed in G if H is strongly closed in N G (H) with respect to G. Let H be a subgroup of a finite group G , then H is called Hall s -semiembedded in G if H is a Hall subgroup of 〈 H , P 〉 for every P ∈ Syl p (G) , where (| H | , p) = 1. In this paper, we obtain some criteria for p -nilpotency of a finite group and extend some known results concerning strongly closed and Hall s -semiembedded subgroups. In particular, we generalize some main results of Guo and Li [Hall s -semiembedded subgroups and p -nilpotency of finite groups, Southeast Asian Bull. Math. 42(3) (2018) 367–374] and Kong [New characterizations of p -nilpotency of finite groups, J. Algebra Appl. 20(11) (2021) Article ID: 2150215, 6 pp.]. [ABSTRACT FROM AUTHOR]

Let J (R) denote the Jacobson radical of a commutative ring R. In [H. A. Khashan and A. B. Bani-Ata, J -ideals of commutative rings, Int. Electron. J. Algebra 29 (2021) 148–164], the notion of J-ideals was introduced. If N (R) denotes the prime radical of a commutative ring then in [U. Tekir, S. Koc and K. H. Oral, n -Ideals of commutative rings, Filomat 31(10) (2017) 2933–2941], the notion of an N ideal of a commutative ring was introduced and studied. In this note, we show that these results are special cases of a more general situation. We define ρ -ideals for a special radical ρ and prove that most of the results of the above-mentioned papers are satisfied for non-commutative rings as a special case. [ABSTRACT FROM AUTHOR]

In [Finite solvable groups with exactly two nonlinear non-faithful irreducible characters, J. Algebra Appl. 18(5) (2019) 1950091], the first author studied directly indecomposable, solvable groups with exactly two nonlinear non-faithful irreducible characters. In this paper, we study the decomposable case and give a classification of decomposable groups possessing exactly two nonlinear non-faithful irreducible characters. In particular, nilpotent groups with this property are also classified. [ABSTRACT FROM AUTHOR]

Let (R ,) be a Noetherian local ring and M a finitely generated R -module. In this paper, we study the polynomial type and the sequential polynomial type of the idealization R ⋉ M of R and M introduced by Cuong [On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings, Nagoya Math. J. 125 (1992) 105–114] and Nhan-Dung-Chau [A measure for non-sequential Cohen Macaulayness of finitely generated modules, J. Algebra 468 (2016) 275–295], respectively. In addition, we give a uniform bound for the reducibility index of parameter ideals of idealizations whose polynomial type and sequential polynomial type are at most one. [ABSTRACT FROM AUTHOR]

The concepts of derivation and centroid for Lie–Yamaguti algebras are generalized in this paper. A quasi-derivation of a LY-algebra can be embedded as a derivation in a larger LY-algebra. The relationship between quasi-derivations and robustness of Lie–Yamaguti algebras has been studied. [ABSTRACT FROM AUTHOR]

In a paper from 2002, Zhang [Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002) 4131–4151] used module extensions to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. Following his approach, we show that for amalgamated duplication of dual Banach algebras, we are able to choose appropriate conditions for weak Connes amenability of those Banach algebras which extend the Zhang's result in a much more general setting. Moreover, apart from the characterization of minimal idempotents, we restrict our investigation to the study of injectivity, projectivity and flatness of amalgamated duplication of Banach algebras. Particularly, it is worth mentioning that our result paves the way for studying of homological properties and weak Connes amenability of those dual Banach algebras that have a semidirect product-like structure including Lau algebras. [ABSTRACT FROM AUTHOR]

Let R be a prime ring, let L be a noncentral Lie ideal of R and let g , h be two generalized derivations of R. In this paper, we characterize the structure of R and all possible forms of g and h such that [ g (x m) x n − x s h (x t) , x r ] k = 0 for all x ∈ L , where m , n , s , t , r , k are fixed positive integers. With this, several known results can be either deduced or generalized. In particular, we give a Lie ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317] and describe a more complete version of the theorem recently obtained by Dhara and De Filippis in [Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48 (2020) 154–167]. [ABSTRACT FROM AUTHOR]

Let Γ be a torsionless grading monoid, R = ⊕ α ∈ Γ R α a Γ -graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of R 0 and G 0 the group of units of Γ. We say that R is a graded pseudo-valuation domain (gr-PVD) if whenever a homogeneous prime ideal P of R contains the product xy of two homogeneous elements of R H , then x ∈ P or y ∈ P. The notion of gr-PVDs was introduced recently by the authors in (M. T. Ahmed, C. Bakkari, N. Mahdou and A. Riffi, Graded pseudo-valuation domains, Comm. Algebra 48 (2020) 4555–4568) as a graded version of pseudo-valuation domains (PVDs). In this paper, we show that R is a gr-PVD if and only if exactly one of the following two conditions holds : (1)(a) R 0 = K , (b) Γ is a pseudo-valuation monoid, and (c) R α = (R H) 0 x for every 0 ≠ x ∈ R α whenever α ∈ Γ is not a unit. (2)(a) R 0 ≠ K , (b) Γ is a valuation monoid, (c) R α = K x for every 0 ≠ x ∈ R α whenever α ∈ Γ is not a unit, and (d) T = ⊕ α ∈ G 0 R α is a gr-PVD. [ABSTRACT FROM AUTHOR]

The aim of this paper is the characterization of algebraic properties of Leavitt path algebra of the directed power graph (G) and also of the directed punctured power graph ∗ (G) of a finite group G. We show that Leavitt path algebra of the power graph (G) of finite group G over a field K is simple if and only if G is a direct sum of finitely many cyclic groups of order 2. Finally, we prove that the Leavitt path algebra L K ( ∗ (G)) is a prime ring if and only if G is either cyclic p -group or generalized quaternion 2 -group. [ABSTRACT FROM AUTHOR]

In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter q and two polynomials f , g. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties. [ABSTRACT FROM AUTHOR]

In this paper, we extend work from [M. Vancliff and P. P. Veerapen, Generalizing the notion of rank to noncommutative quadratic forms, in Noncommutative Birational Geometry, Representations and Combinatorics, eds. A. Berenstein and V. Retakh, Contemporary Mathematics, Vol. 592 (2013), pp. 241–250], where a notion of rank, called μ -rank, was proposed for noncommutative quadratic forms on two and three generators. In particular, we provide a definition of μ -rank one and two for noncommutative quadratic forms on four generators. We apply this definition to determine the number of point modules over certain quadratic AS-regular algebras of global dimension four. [ABSTRACT FROM AUTHOR]

In this paper, we study the irreducible quotient ℒ t , c of the polynomial representation of the rational Cherednik algebra ℋ t , c (S n ,) of type A n − 1 over an algebraically closed field of positive characteristic p where p | n − 1. In the t = 0 case, for all c ≠ 0 we give a complete description of the polynomials in the maximal proper graded submodule ker ℬ , the kernel of the contravariant form ℬ , and subsequently find the Hilbert series of the irreducible quotient ℒ 0 , c . In the t = 1 case, we give a complete description of the polynomials in ker ℬ when the characteristic p = 2 and c is transcendental over 2 , and compute the Hilbert series of the irreducible quotient ℒ 1 , c . In doing so, we prove a conjecture due to Etingof and Rains completely for p = 2 , and also for any t = 0 and n ≡ 1 (mod p). Furthermore, for t = 1 , we prove a simple criterion to determine whether a given polynomial f lies in ker ℬ for all n = k p + r with r and p fixed. [ABSTRACT FROM AUTHOR]

We present an answer to a problem raised by Anderson and Chun in [D. D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16(7) (2017) 1750143] on characterizing when the idealization R ⋉ M of a ring R on an R -module M is an 𝒜 -ring (respectively, an 𝒮 𝒜 -ring) in terms of module-theoretic properties of R and M. Also, we are concerned with an open question asked by these two authors which reads the following: What modules over a given ring R are homomorphic images of modules satisfying the strong Property 𝒜 ? (see, Question 4.4(1) in the above mentioned paper). This paper highly contributes to answer such a question. [ABSTRACT FROM AUTHOR]

The notion of f -ideal was introduced in [G. Q. Abbasi, S. Ahmad, I. Anwar and W. A. Baig, f -Ideals of degree 2, Algebra Colloq.19(1) (2012) 921–926] in 2012 and has been studied in many papers after that. In this paper, we have studied those graphs whose Stanley–Reisner ideals turn out to be f -ideals. We give a characterization and construction of these graphs and show that, unlike f -graphs, these graphs are always connected. We have also discussed when these graphs are complete bipartite graphs. Moreover, we classify those graphs for which both the facet ideal (the edge ideal) and the Stanley–Resiner ideal, are f -ideals. [ABSTRACT FROM AUTHOR]

The aim of the paper is to prove two conjectures from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra 𝕀 n : = K 〈 x 1 , ... , x n , ∂ ∂ x 1 , ... , ∂ ∂ x n , ∫ 1 , ... , ∫ n 〉 of polynomial integro-differential operators and the Jacobian algebra 𝔸 n is equal to n (over a field of characteristic zero). The algebras 𝕀 n and 𝔸 n are neither left nor right Noetherian and 𝕀 n ⊂ 𝔸 n . Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert's Syzygy Theorem is proven for the algebras 𝕀 n , 𝔸 n and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras 𝕀 n and 𝔸 n is n and the weak global dimension of all the factor algebras of 𝕀 n and 𝔸 n is n. [ABSTRACT FROM AUTHOR]

In this paper, we consider the notion of semiprime submodules, which is a natural generation of semiprime ideals. The main purpose of this paper is to give local-global properties of semiprimes and semiprime radicals as well as results on finitely generated multiplication modules. Also, we prove that for every submodule N of a module M , the semiprime radical of N in M is the direct limit of the ascending chain E M (N) ⊆ E M (E M (N)) ⊆ ⋯ of submodules of M , and we prove that N satisfies the semiprime radical formula if and only if its envelope E M (N) in M is semiprime. [ABSTRACT FROM AUTHOR]

In this paper, we compute the dimension of the Hochschild cohomology groups of any m -cluster tilted algebra of type 𝔸 ̃. Moreover, we give conditions on the bounded quiver of an m -cluster tilted algebra Λ of type 𝔸 ̃ such that the Gerstenhaber algebra HH ∗ (Λ) has nontrivial multiplicative structures. We also show that the derived class of gentle m -cluster tilted algebras is not always completely determined by the dimension of the Hochschild cohomology. [ABSTRACT FROM AUTHOR]

In this paper, we study some algebras A whose unit groups 𝒰 (A) or subnormal subgroups of 𝒰 (A) are (generalized) Engel. For example, we show that any generalized Engel subnormal subgroup of the multiplicative group of division rings with uncountable centers is central. Some of algebraic structures of Engel subnormal subgroups of the unit groups of skew group algebras over locally finite or torsion groups are also investigated. [ABSTRACT FROM AUTHOR]

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils et al. and we call the restricted Jordan plane. In this paper, the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are classified. The defining relations of the graded dual of the Jordan plane are given. [ABSTRACT FROM AUTHOR]

The paper is devoted to study nilpotent Jordan algebras 𝒥 satisfying that 𝒥 / Ann (𝒥) is non-associative. We show that such algebras exist iff dim 𝒥 ≥ 6 and dim (𝒥 / Ann (𝒥)) ≥ 5. Also we classify, up to isomorphism, all 6 -dimensional nilpotent Jordan algebras 𝒥 such that 𝒥 / Ann (𝒥) is non-associative, over an algebraically closed field 𝔽 of characteristic ≠ 2 , 3. [ABSTRACT FROM AUTHOR]

Let f : A → B be a ring homomorphism, J be an ideal of B and A ⋈ f J the amalgamation of A with B along J with respect to f. In this paper, we provide necessary and sufficient conditions for A ⋈ f J to be a Steinitz ring, semi-Steinitz ring, and weakly semi-Steinitz. Then we construct new original examples of weakly semi-Steinitz rings which are not semi-Steinitz rings and semi-Steinitz rings which are not Steinitz rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce weak (Σ , Δ) -compatible rings and study skew Poincaré–Birkhoff–Witt extensions over these rings. We characterize the weak notion of compatibility for several noncommutative rings appearing in noncommutative algebraic geometry and some quantum algebras of theoretical physics. As a consequence of our treatment, we unify and extend results in the literature about Ore extensions and skew PBW extensions over compatible rings. [ABSTRACT FROM AUTHOR]

Let K be a field with char (K) = 0 and E = K 〈 e 1 , ... , e n 〉 an exterior algebra over K with a standard grading deg e i = 1. Let R = E / J be a graded algebra, where J is a graded ideal in E. In this paper, we study universally Koszul and initially Koszul properties of R and find classes of ideals J which characterize such properties of R. As applications, we classify arrangements whose Orlik–Solomon algebras are universally Koszul or initially Koszul. These results are related to a long-standing question of Shelton–Yuzvinsky [B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc.56 (1997) 477–490]. [ABSTRACT FROM AUTHOR]

In this paper, we propose a graph superalgebra which is the supersymmetric analogue of Leavitt path algebras. We find a basis for these superalgebras and characterize when they have polynomial growth. [ABSTRACT FROM AUTHOR]

The aim of the paper is to extend the class of generalized Weyl algebras (GWAs) to a larger class of rings (they are also called GWAs) that are determined by two ring endomorphisms rather than one as in the case of 'old' GWAs. A new class of rings, the diskew polynomial rings, is introduced that is closely related to GWAs (they are GWAs under a mild condition). Simplicity criteria are given for GWAs and diskew polynomial rings. [ABSTRACT FROM AUTHOR]

Given a directed graph E and an associative unital ring R one may define the Leavitt path algebra with coefficients in R , denoted by L R (E). For an arbitrary group G , L R (E) can be viewed as a G -graded ring. In this paper, we show that L R (E) is always nearly epsilon-strongly G -graded. We also show that if E is finite, then L R (E) is epsilon-strongly G -graded. We present a new proof of Hazrat's characterization of strongly ℤ -graded Leavitt path algebras, when E is finite. Moreover, if E is row-finite and has no source, then we show that L R (E) is strongly ℤ -graded if and only if E has no sink. We also use a result concerning Frobenius epsilon-strongly G -graded rings, where G is finite, to obtain criteria which ensure that L R (E) is Frobenius over its identity component. [ABSTRACT FROM AUTHOR]

Generalizing the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC), a class of constraints involving a pair of operators was considered in [Extensions of operators, liftings of monads, and mixed distributive laws, Appl. Categ. Struct. 26 (2018) 747–765]. For a given constraint, the existences of extensions of differential and Rota–Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are shown to be equivalent. In this paper, we give a classification of the constraints satisfying these equivalent conditions. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities (ADE) is of linear type. We show that any reduced projective quartic curve is of gradient linear type. [ABSTRACT FROM AUTHOR]

In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph homomorphisms, and obtain a covariant functor from a graph category to an algebra category. [ABSTRACT FROM AUTHOR]

In this paper, we define and study weak monoidal Hom-Hopf algebras, which generalize both weak Hopf algebras and monoidal Hom-Hopf algebras. Let H be a weak monoidal Hom-Hopf algebra with bijective antipode and let Aut w m H H (H) be the set of all automorphisms of H , we introduce a category H 𝒲 ℳ ℋ 𝒴 𝒟 H (α , β) with α , β ∈ Aut w m H H (H) and construct a braided T -category 𝒲 ℳ ℋ 𝒴 𝒟 (H) having all the categories H 𝒲 ℳ ℋ 𝒴 𝒟 H (α , β) as components. [ABSTRACT FROM AUTHOR]

In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of regular Hom-pre-Lie algebras in terms of the cohomology theory of regular Hom-Lie algebras. As applications, we study linear deformations of regular Hom-pre-Lie algebras, which are characterized by the second cohomology groups of regular Hom-pre-Lie algebras with the coefficients in the regular representations. The notion of a Nijenhuis operator on a regular Hom-pre-Lie algebra is introduced which can generate a trivial linear deformation of a regular Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a regular Hom-pre-Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an 𝒪 -operator on a regular Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure. [ABSTRACT FROM AUTHOR]

In this paper, we study (asymptotic) properties of the ∗-distribution of irreducible characters of finite quantum groups. We proceed in two steps, first examining the representation theory to determine irreducible representations and their powers, then we study the ∗-distribution of their trace with respect to the Haar measure. For the Sekine family, we look at the asymptotic distribution (as the dimension of the algebra goes to infinity). [ABSTRACT FROM AUTHOR]

Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra L , in spite of being noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [The multiplicative ideal theory of Leavitt path algebras, J. Algebra 487 (2017) 173–199], it was shown that the ideals of L satisfy the distributive law, a property of Prüfer domains and that L is a multiplication ring, a property of Dedekind domains. In this paper, we first show that L satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers a , b , c , gcd (a , b) ⋅ lcm (a , b) = a ⋅ b and a ⋅ gcd (b , c) = gcd (a b , a c). We also show that L satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which L satisfies another important characterizing property of almost Dedekind domains, namely, the cancellative property of its nonzero ideals. [ABSTRACT FROM AUTHOR]

In this paper, we present a complete method for finding the roots of all polynomials of the form ϕ (z) = c n z n + c n − 1 z n − 1 + ⋯ + c 1 z + c 0 over a given octonion division algebra. When ϕ (z) is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of ϕ (z) , showing that the right eigenvalues form the conjugacy classes of the roots of ϕ (z) and the left eigenvalues form a larger set than the roots of ϕ (z). [ABSTRACT FROM AUTHOR]

In this paper, we give the oriented quantum algebra (OQA) structures on the tensor product of two different OQAs by using Chen's weak ℜ -matrix in [J. Algebra204 (1998) 504–531]. As a special case, the OQA structures on the tensor product of an OQA with itself are provided, which are different from Radford's results in [J. Knot Theory Ramifications16 (2007) 929–957]. [ABSTRACT FROM AUTHOR]

In the present paper, we study 2-local derivations of infinite-dimensional Lie algebras over a field of characteristic zero. We prove that all 2-local derivations of the Witt algebra as well as of the positive Witt algebra and the classical one-sided Witt algebra are (global) derivations. We also give an example of an infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation. [ABSTRACT FROM AUTHOR]

Let K [ X d ] = K [ x 1 , ... , x d ] be the polynomial algebra in d variables over a field K of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations δ (known as Weitzenböck derivations), the algebra of constants K [ X d ] δ is finitely generated. When the Weitzenböck derivation δ acts on the polynomial algebra K [ X d , Y d ] in 2 d variables by δ (y i) = x i , δ (x i) = 0 , i = 1 , ... , d , Nowicki conjectured that K [ X d , Y d ] δ is generated by X d and x i y j − y i x j for all 1 ≤ i < j ≤ d. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free d -generated metabelian Lie algebra F d , with few trivial exceptions, the algebra F d δ is not finitely generated. However, the vector subspace (F d ′) δ of the commutator ideal F d ′ of F d is finitely generated as a K [ X d ] δ -module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the K [ X d , Y d ] δ -module (F 2 d ′) δ . [ABSTRACT FROM AUTHOR]

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra H , which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of H -modules is equivalent to the category H H 𝒬 𝒴 𝒟 of quasi-Yetter–Drinfeld modules. Next, we prove that H H 𝒬 𝒴 𝒟 is equivalent to the category of generalized Hopf bimodules. Finally, we show that H H 𝒬 𝒴 𝒟 is also equivalent to the category of quasi-coactions over some Majid's braided group if H is quasi-triangular. [ABSTRACT FROM AUTHOR]

In this paper, we give a recursive formula for the interesting PBW basis E A of composition subalgebras ℭ ▵ (n) of Ringel–Hall algebras ℌ ▵ (n) of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of U v + ( 𝔰 𝔩 ̂ n) from the monomial bases m (A) following [Multiplication formulas and canonical basis for quantum affine, 𝔤 𝔩 n , Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of U v + ( 𝔰 𝔩 ̂ 2) associated with modules of Loewy length ≤ 3. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras. [ABSTRACT FROM AUTHOR]

This paper belongs to the series of articles devoted to the classification of deformed mesh algebras of Dynkin types 𝔸 n (n ≥ 2) , 𝔹 n (n ≥ 2) , ℂ n (n ≥ 3) , 𝔻 n (n ≥ 4) , 𝔼 6 , 𝔼 7 , 𝔼 8 , 𝔽 4 , 𝔾 2 , and 𝕃 n (n ≥ 1). We prove here that every deformed mesh algebra of type 𝔽 4 is isomorphic to the canonical mesh algebra of type 𝔽 4 . [ABSTRACT FROM AUTHOR]

In this note, we use the theory of test ideals and Cartier algebras to examine the interplay between the tight and integral closures in a local ring of positive characteristic. Using work of Schwede, we prove the abundance of strong test ideals, recovering some older fundamental results, and use this approach in concrete computations. In the second part of the paper, the case of Stanley–Reisner rings is fully examined. [ABSTRACT FROM AUTHOR]

The power graph 𝒫 (G) of a given finite group G is the simple undirected graph whose vertices are the elements of G , in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity κ (𝒫 (G)) of 𝒫 (G) is the minimum number of vertices which need to be removed from G so that the induced subgraph of 𝒫 (G) on the remaining vertices is disconnected or has only one vertex. For a positive integer n , let C n be the cyclic group of order n. Suppose that the prime power decomposition of n is given by n = p 1 n 1 p 2 n 2 ⋯ p r n r , where r ≥ 1 , n 1 , n 2 , ... , n r are positive integers and p 1 , p 2 , ... , p r are prime numbers with p 1 < p 2 < ⋯ < p r . The vertex connectivity κ (𝒫 (C n)) of 𝒫 (C n) is known for r ≤ 3 , see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl.17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for r ≥ 4 , we give a new upper bound for κ (𝒫 (C n)) and determine κ (𝒫 (C n)) when n r ≥ 2. We also determine κ (𝒫 (C n)) when n is a product of distinct prime numbers. [ABSTRACT FROM AUTHOR]

It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper, we establish such a generalization of the "salamander lemma" due to G. M. Bergman, in a self-dual axiomatic context (developed originally by Z. Janelidze), which applies to all usual non-abelian group-like structures and also covers axiomatic contexts such as semi-abelian categories in the sense of G. Janelidze, L. Márki and W. Tholen and exact categories in the sense of M. Grandis. [ABSTRACT FROM AUTHOR]

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of the notion of support of a hypermodule over a Krasner hyperring, providing several connections with the annihilator of such hypermodules. We concentrate on the similarities/differences between these concepts and the analogous ones from classical algebra (module theory). After defining and characterizing the support of a hypermodule (and in particular of a finitely generated hypermodule), by using the notion of length of a hypermodule, we determine the support of a quotient hypermodule containing only one maximal hyperideal. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of n - L i e -isoclinism on non-Lie Leibniz algebras. Among the results obtained, we provide several characterizations of n - L i e -isoclinic classes of Leibniz algebras. Also, we provide a characterization of n - L i e -stem Leibniz algebras, and prove that every n - L i e -isoclinic class of Leibniz algebras contains a n - L i e -stem Leibniz algebra. [ABSTRACT FROM AUTHOR]