In this work, an algebraic method to prove the existence of left eigenvalues for the quaternionic matrix is investigated. The left eigenvalues of a n×n quaternionic matrix can be derived by solving the zeros of a general quaternionic polynomial of degree 2n. Using the Study’s determinant, it can be found by solving the zeros of quaternionic polynomials of degree at most n or of rational functions.

Let R be a commutative ring with 1≠0. A proper ideal I of R is said to be a strongly quasi-primary ideal if, whenever a,b∈R with ab∈I, then either a2∈I or b∈I. In this paper, we characterize Noetherian and reduced rings over which every (respectively, nonzero) proper ideal is strongly quasi-primary. We also characterize ring over which every strongly quasi primary ideal of R is prime. Many examples

The zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices are the nonzero zero divisors in R and two vertices x and y are adjacent if and only if xy=0. We study the adjacency and Laplacian eigenvalues of the zero-divisor graph Γ(R) of a finite commutative von Neumann regular ring R. We prove that Γ(R) is a generalized join of its induced subgraphs. Among the |Z(R)∗| eigenvalues

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of W(n) under some conditions

Let R be a commutative ring with unity, a,b∈R, and I an ideal of R. Define R(I)a,b to be ℛ+/(I2(t2+at+b)), a quotient of the Rees algebra. In this paper, we investigate when the rings in the family are generalized Cohen–Macaulay or filter rings and show that these properties are independent of the choice of a and b.

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [d,1] to be a subalgebra of an involutive residuated lattice, where d is the dualizing element. We also import some results and techniques of semimodule

Let R=AMNB be a Morita context. For generalized fine (respectively, generalized unit-fine) rings A and B, it is proved that R is generalized fine (respectively, generalized unit-fine) if and only if, for a∈A and b∈B, MbN⊆J(A) implies b∈J(B) and NaM⊆J(B) implies a∈J(A). Especially, for fine (respectively, unit-fine) rings A and B, R is fine (respectively, unit-fine) if and only if, for a∈A and b∈B,

Let R be a prime ring with the extended centroid C and L a noncentral Lie ideal of R. Suppose that a∈R, γ∈C and d is a derivation of R such that ad(u2)s−γu2tm∈C for all u∈L, where s,t,m are fixed positive integers. If γ≠0 or d≠0, then a=0 unless R⊆M2(F), the 2×2 matrix ring over a field F. This result gives an affirmative answer to the open conjecture recently raised by Huang in [Derivations with annihilator

An irreducible character of a finite group G is called quasi p-Steinberg character for a prime p if it takes a nonzero value on every p-regular element of G. In this paper, we classify the quasi p-Steinberg characters of Symmetric (Sn) and Alternating (An) groups and their double covers. In particular, an existence of a nonlinear quasi p-Steinberg character of Sn implies n≤8 and of An implies n≤9.

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

Let 𝒜 be an abelian category. In this paper, we investigate the global (𝒳,𝒴)-Gorenstein projective dimension gl.GPD(𝒳,𝒴)(𝒜), associated to a GP-admissible pair (𝒳,𝒴). We give homological conditions over (𝒳,𝒴) that characterize it. Moreover, given a GI-admissible pair (𝒵,𝒲), we study conditions under which gl.GID(𝒵,𝒲)(𝒜) and gl.GPD(𝒳,𝒴)(𝒜) are the same.

The aim of this paper is the characterization of algebraic properties of Leavitt path algebra of the directed power graph 𝒫(G) and also of the directed punctured power graph 𝒫∗(G) of a finite group G. We show that Leavitt path algebra of the power graph 𝒫(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

Let R be a commutative ring. If the nilpotent radical Nil(R) of R is a divided prime ideal, then R is called a ϕ-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and ϕ-coherent rings introduced by Bacem and Ali [Nonnil-coherent rings, Beitr. Algebra Geom.57(2) (2016) 297–305], and then characterize nonnil-coherent rings in terms of ϕ-flat modules, nonnil-injective modules

The zero divisor graph Γ(R) of a commutative ring R with unity is a simple undirected graph whose vertices are all nonzero zero divisors of R and two distinct vertices x and y are adjacent if and only if xy=0. In this paper, we study the graphical structure and the adjacency spectrum of the zero divisor graph of ring ℤn. For any non-prime positive integer n≥4 with ξ number of proper divisors, we show

Let G be a finite non-abelian group and let T be a transversal of the center of G. The non-commuting graph of G on a transversal of the center is the graph denoted by 𝕋(G) whose vertices are the non-central elements of T and two vertices x and y are joined by an edge whenever xy≠yx. In this paper, we determine (up to isoclinism) all finite non-abelian groups whose non-commuting graph on a transversal

In the hyperalgebra 𝒰r of the rth Frobenius kernel (SL2)r of the algebraic group SL2, we construct a basis of the 𝒰r-module generated by a certain element which was given by the author before. As its applications, we also prove some results on the 𝒰r-modules and the algebra 𝒰r.

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive

Let R be a commutative ring with identity, S be a multiplicatively closed subset of R, and M be an R-module. A submodule N of M is called coidempotent if N=((0:MAnnR2(N)). Also, M is called fully coidempotent if every submodule of M is coidempotent. In this paper, we introduce the concepts of S-coidempotent submodules and fully S-coidempotent R-modules as generalizations of coidempotent submodules

We provide necessary and sufficient conditions on the graph E and the field K for which the Leavitt path algebra LK(E) is Lie solvable. Consequently, we obtain a complete description of Lie nilpotent Leavitt path algebras, and show that the Lie solvability of LK(E) and the Lie nilpotency of [LK(E),LK(E)] are the same. Furthermore, we compute the solvable index of a Lie solvable Leavitt path algebra

In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in terms of associated combinatorial data and show that the lower bound is attained in some cases. Also, we obtain bounds on the projective dimension of edge ideals of

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

Given an affine algebra R=P/I, where P=K[x1,…,xn] is a polynomial ring over a field K and I is an ideal in P, we study re-embeddings of the affine scheme Spec(R), i.e. presentations R≅P′/I′ such that P′ is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials fi in the ideal I which are coherently separating in the sense that they are of the form fi=zi−gi with an

Let G be a finite group and H a subgroup of G. H is said to be NH-embedded in G if there exists a normal subgroup T of G such that HT is a Hall subgroup of G and H∩T≤Hs̄G, where Hs̄G is the largest s-semipermutable subgroup of G contained in H. In this paper, we give some new characterizations of p-nilpotent and supersolvable groups by using NH-embedded subgroups. Some known results are generalized

In this paper, we have introduced copure-direct-injective modules. A right R-module M is said to be copure-direct-injective if every copure submodule of M isomorphic to a direct summand of M is itself a direct summand. We have studied properties of copure-direct-injective modules. We characterized rings over which every (cofinitely generated, free, projective) module is copure-direct-injective. We

In this paper, we introduce the notions of Sϕ-polynomial and S-minimal value set polynomial where ϕ is a polynomial over a finite field 𝔽q and S is a finite subset of an algebraic closure of 𝔽q. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are Sϕ-minimal value set polynomials for

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit

In this paper, we study the irreducible quotient ℒt,c of the polynomial representation of the rational Cherednik algebra ℋt,c(Sn,𝔥) of type An−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 kerℬ, the kernel of the contravariant form ℬ, and subsequently

Let 𝒜 be an abelian category and 𝒳,𝒞 be two quasi-resolving subcategories of 𝒜, where 𝒞 is closed under direct summands and 𝒞 is a cogenerator for 𝒳. For any M∈𝒳⊥1, it is shown that 𝒳-pd(M)=𝒞-pd(M). Some examples and applications are also given.

In this paper, we introduce and study the concept of strongly dccr⋆ modules. Strongly dccr⋆ condition generalizes the class of Artinian modules and it is stronger than dccr⋆ condition. Let R be a commutative ring with nonzero identity and M a unital R-module. A module M is said to be strongly dccr⋆ if for every submodule N of M and every sequence of elements (ai) of R, the descending chain of submodules

Wu–Liu–Ding algebras are a class of affine prime regular Hopf algebras of GK-dimension one, denoted by D(m,d,ξ). In this paper, we consider their quotient algebras D′(m,d,ξ), which are a new class of non-pointed semisimple Hopf algebras. We describe the Grothendieck rings of D′(m,d,ξ) when d is odd. It turns out that the Grothendieck rings are commutative rings generated by three elements subject to

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic

A theorem of Lichtenbaum states, that every proper, regular curve 𝒞 over a discrete valuation domain R is projective. This theorem is generalized to the case of an arbitrary valuation domain R using the following notion of regularity for non-noetherian rings introduced by Bertin: the local ring 𝒪𝒞,x of a point x∈𝒞 is called regular, if every finitely generated ideal I has finite projective dimension

Analogy between Anderson t-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: (1) A description of abelian varieties with MIQF of dimension r and signature (n,r−n) in terms of ”lattices” of dimension r in ℂn; (2) A construction of exterior powers of abelian varieties with MIQF having n=1.

In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J.61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we

It is shown that every finitely generated right R-module is almost injective if and only if every cyclic right R-module is almost injective, if and only if R/J(R) is a right SV-ring with Loewy(RR)≤2 and there is a finite set of orthogonal idempotents {ei}i∈I in R such that eiR is an injective local right R-module of length two for every i∈I and J(R)=⊕i∈IJ(eiR).

Let R be a commutative ring and let N be a submodule of Rn which consists of columns of a matrix A=(aij) with aij∈R for all 1≤i≤n, j∈Λ, where Λ is an index set. For every μ={j1,…,jq}⊆Λ, let Iμ(N) be the ideal generated by subdeterminants of size q of the matrix (aij:1≤i≤n,j∈μ). Let M=Rn/N. In this paper, we obtain a constructive description of T(M) and we show that when R is a local ring, M/T(M) is

We study stability conditions on the Calabi–Yau-N categories associated to an affine type An quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order N−2. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show

In this paper, we will introduce the concept of sympathetic 3-Lie algebras and show that some classical properties of semi-simple 3-Lie algebras are still valid for sympathetic 3-Lie algebras. We prove that every perfect 3-Lie algebra L contains a greatest special sympathetic ideal M, and there exists a solvable ideal of L denoted by P(L) which is the greatest among the solvable ideals I of L for which

Let R be a commutative ring with identity, X be an indeterminate and Ic(R) be the set of elements a of R such that there exists an R-homomorphism of rings σ:R[[X]]→R with σ(X)=a. O’Malley called R to be power invariant (respectively, strongly power invariant) if whenever S is a ring such that R[[X]] is isomorphic to S[[X]] (respectively, whenever S is a ring and φ is an isomorphism of R[[X]] onto S[[X]])

We adapt and generalize results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this sequence leads to a classification of extensions by a second cohomology group. Our methods use structural ideas in cohomology as far as possible, rather than computation

In this paper, we introduce the notions of non-abelian tensor and exterior products of two ideal graded crossed submodules of a given crossed module of Lie superalgebras. We also study some of their basic properties and their connection with the second homology of crossed modules of Lie superalgebras.

We study the absolute-valued algebras 𝒜 satisfying the identity (xx2)x=x(x2x). We show that, if moreover, the norm of 𝒜 comes from an inner product and 𝒜 contains a nonzero central element then 𝒜 is finite dimensional and isomorphic to either ℝ,ℂ,ℂ∗,ℍ,ℍ∗,𝕆, or 𝕆∗.

We present an elementary approach to characterizing Lie polynomials on the generators A,B of an algebra with a defining relation in the form of a twisted commutation relation AB=σ(BA). Here, the twisting map σ is a linear polynomial with a slope parameter, which is not a root of unity. The class of algebras defined as such encompasses q-deformed Heisenberg algebras, rotation algebras, and some types

Let D be an integral domain with quotient field K, throughout. Call two elements x,y∈D∖{0}v-coprime if xD∩yD=xyD. Call a nonzero non-unit r of an integral domain D rigid if for all x,y|r we have x|y or y|x. Also, call D semirigid if every nonzero non-unit of D is expressible as a finite product of rigid elements. We show that a semirigid domain D is a GCD domain if and only if D satisfies ∗: product

Building on recent work of Jaikin-Zapirain, we provide a homological criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a division ring of fractions over which all stably full matrices become invertible. We use the criterion to study skew Laurent polynomial rings over free ideal rings (firs). As an application of our methods, we prove that crossed products of division rings with

Let G be a group. The enhanced power graph of G is a graph with vertex set G and two distinct vertices x and y are adjacent if there exists z∈G such that x=zm and y=zn for some m,n∈ℕ. Also, a vertex of a graph is called dominating vertex if it is adjacent to every other vertex of the vertex set. Moreover, an enhanced power graph is said to be a dominatable graph if it has a dominating vertex other

In this paper, we characterize the (covariant) isotropy groups of free, finitely generated racks and quandles. As a consequence, we show that the usual inner automorphisms of such racks and quandles are precisely those automorphisms that are “coherently extendible”. We then use this result to compute the global isotropy groups of the categories of racks and quandles, i.e. the automorphism groups of

Let 𝒰 be a 2-torsion free unital generalized matrix algebra, and ϕ:𝒰→𝒰 be a linear map satisfying S,T∈𝒰,ST=0⇒ϕ([S,T])=[ϕ(S),T]=[S,ϕ(T)]. In this paper, we study the structure of ϕ and under some mild conditions on 𝒰 we present the necessary and sufficient conditions for ϕ to be in terms of centralizers. We then provide the characterizations of Lie centralizers on 𝒰 and our results generalize

Let R be a commutative local ring whose maximal ideal is generated by a nilpotent element, and Mat(n,R) be the multiplicative monoid of the square matrices of order n over R. In this paper, we provide the construction of Green’s ℋ-equivalence classes in the multiplicative monoid Mat(n,R). Then, we enumerate these classes in the special cases R=ℤ/pdℤ and R=𝔽q[x]/〈xd〉.

Let R be a ring. We study the FG-pure exact sequences, which are defined analogously to pure exact sequences except with respect to finitely generated R-modules rather than the finitely presented R-modules. For Noetherian rings the two notions coincide. But for non-Noetherian rings, the FG-purity is sharper than the usual purity. We also study the corresponding notion of FG-flat module and construct

In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties under homomorphic image and their transfer to various contexts of constructions such as direct product, trivial ring extension and amalgamated duplication of a ring

Rings in which each finitely generated right ideal is automorphism-invariant (rightfa-rings) are shown to be isomorphic to a formal matrix ring. Among other results it is also shown that (i) if R is a right nonsingular ring and n>1 is an integer, then R is a right self injective regular ring if and only if the matrix ring 𝕄n(R) is a right fa-ring, if and only if 𝕄n(R) is a right automorphism-invariant

In 2004, Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a ℤ-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that cannot be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms.

We study canonical bases of a subalgebra A=𝕂[[f1,…,fs]]⊆𝕂[[x1,…,xn]] over a field 𝕂, and we associate with A a fan called the canonical fan of A. This generalizes the notion of the standard fan of an ideal.

In this paper, we mainly investigate the class-preserving Coleman automorphisms of finite groups whose Sylow 2-subgroups are semidihedral. We prove that if G is a finite solvable group with semidihedral Sylow 2-subgroups, then Outc(G)∩OutCol(G) is a 2′-group and therefore G satisfies the normalizer property. As some applications of this result, we also investigate the normalizer property of the following

We study some properties of formal local cohomology modules ℱ𝔞i(M) in Serre subcategories. As a consequence, we obtain some results on the minimax modules. We also describe the sets CosR(ℱ𝔞i(M)) and CoassR(ℱ𝔞i(M)).

A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang–Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions obtained by Rump in A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193 (2005) 40–55, https://doi.org/10

The Bochner–Schoenberg–Eberlein (BSE)-property on commutative Banach algebras is a property related to multiplier algebras of Banach algebras. In this paper, we answer the problem (12) raised by Javanshiri and Nemati in [Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜, J. Algebra Appl.17(9) (2018) 1850169-1–1850169-21]. In this paper, under certain conditions, we show that the

In this paper, we prove that if R is a local ring of dimension d,d≥2 and 1d!∈R then the group Umd+1(R[X])Ed+1(R[X]) has no k-torsion, provided kR=R. We also prove that if R is a regular ring of dimension d,d≥2 and 1d!∈R such that Ed+1(R) acts transitively on Umd+1(R) then Ed+1(R[X]) acts transitively on Umd+1(R[X]).

In this paper, we study the behavior of endomorphism rings of indecomposable (uniform) quasi-injective modules. A very natural question here is, for a morphism f:A→B, with A,B indecomposable (uniform) quasi-injective right R-modules, and g:E(A)→E(B) an extension of f where E(−) denotes the injective hull, what is the relation between kernels of f and g, their monogeny classes and their upper parts