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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]

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]

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]

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]

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]

In an unpublished work of Fasel–Rao–Swan the notion of the relative Witt group W E (R , I) is defined. In this paper, we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol V R , I : U m 3 (R , I) / E 3 (R , I) → W E (R , I). We establish injectivity of this symbol if R is a non-singular affine algebra of dimension 3 over a perfect C 1 -field and I is a local complete intersection ideal of R. It is natural to ask whether the Vaserstein symbol is injective for a singular 3-dimensional affine algebra. At the end of the paper, we will give an example of a singular 3-dimensional algebra over a perfect C 1 -field for which the Vaserstein symbol is injective. [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]

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]

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]

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]

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 K be a field of characteristic zero and x a free variable. A K - ℰ -derivation of K [ x ] is a K -linear map of the form I , − , ϕ for some K -algebra endomorphism ϕ of K [ x ] , where I denotes the identity map of K [ x ]. In this paper, we study the image of an ideal of K [ x ] under some K -derivations and K - ℰ -derivations of K [ x ]. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all K - ℰ -derivations and all locally finite K -derivations of K [ x ]. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent K -derivations of K [ x ] , and also for all locally nilpotent K - ℰ -derivations of K [ x ] and the ideals u K [ x ] such that either u = 0 , or deg u ≤ 1 , or u has at least one repeated root in the algebraic closure of K. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified. [ABSTRACT FROM AUTHOR]

For an ample groupoid and a unit x of , Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra A R () and the category of modules over the isotropy group algebra R x . In this paper, we prove a graded version of these functors and give a description of spectral graded simple modules over the graded Steinberg algebra A R (). As an application, the spectral simple and graded simple modules over the Leavitt path algebra L K (E) are classified. In particular, we show that many of previously known simple and graded simple L K (E) -modules, including the Chen simple modules, are induced from (ungraded or graded) simple modules over isotropy group algebras. [ABSTRACT FROM AUTHOR]

Given a Hopf algebra H , Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld H -module algebra A gives rise to an associative A -bialgebroid structure on the smash product algebra A ♯ H. They also exhibited an antipode map making A ♯ H the total algebra of a Lu's Hopf algebroid over A. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair A and A op of braided commutative Yetter–Drinfeld H -module algebras, and output is a symmetric Hopf algebroid A ♯ H ≅ H ♯ A op over  A. This construction does not require that the antipode of H is invertible. [ABSTRACT FROM AUTHOR]

Action of a pomonoid on partially ordered sets (S -posets) has beautiful aspects in practical subjects such as automata theory, projection algebra and theoretical computer science which makes it always capture the interest of mathematicians. On the other hand, the study of different kinds of weakly injectivity (which category theory inherited from homological and commutative algebra) is an interesting subject for mathematicians. One of the important kinds of weakly injectivity is quasi-injectivity. In this paper, we study quasi-injectivity in the category of S -posets with respect to special kind of order embeddings, namely, down-closed embeddings (dc-quasi-injectivity). [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of Hurwitz series over a commutative unital ring that was begun in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. In particular, we introduce the notion of multivariate Hurwitz series. The underlying idea is that multivariate Hurwitz series are to Hurwitz series as studied in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] as formal power series in several indeterminates are to formal power series in only one indeterminate. The elementary aspects of the theory follow along the lines of [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. The treatment of substitution and divided powers introduces special problems not encountered in [W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] and requires special attention to subtle details. However, we are able to establish analogous results. With substitution and divided powers in place, we construct and study the analog to the so called inner transformations of [S. Bochner and W. T. Martin, Several Complex Variables (Princeton University Press, 1948)]. Finally, we are able to establish analogs to many of the fundamental results of single and multivariate calculus. [ABSTRACT FROM AUTHOR]

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a λ -TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative λ -TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the λ -TD shuffle product. We then show that the free commutative λ -TD algebra possesses a left counital bialgebra structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative λ -TD algebra is connected and filtered, and thus is a left counital Hopf algebra. [ABSTRACT FROM AUTHOR]

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number ρ (n , m , r) associated with the maximal order in the quaternion algebra (− 1 , − 1) ℚ . As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun's "1-3-5 conjecture". Furthermore, we show that "a perfect square" in the integral condition version of Sun's conjecture can be replaced by "a power of 4". [ABSTRACT FROM AUTHOR]

In this paper, we classify the derivations of degree-one generalized Weyl algebras over a univariate Laurent polynomial ring. In particular, our results cover the Weyl–Hayashi algebra, a quantization of the first Weyl algebra arising as a primitive factor algebra of U q + ( 5) , and a family of algebras which localize to the group algebra of the infinite group with generators x and y , subject to the relation x y = y − 1 x. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if R is an affine algebra of dimension d ≥ 4 over ¯ p and 1 / (d − 1) ! ∈ R , then any unimodular row over R of length d can be mapped to a factorial row by elementary transformations. [ABSTRACT FROM AUTHOR]

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 which is called B -subalgebras (Definition 3.1). We classify the conjugacy classes of B -subalgebras for W (n) and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of W (n). [ABSTRACT FROM AUTHOR]

In this paper, we study properties such as simplicity, solvability and nilpotency for Lie bracket algebras arising from Leavitt path algebras, based on the talented monoid of the underlying graph. We show that graded simplicity and simplicity of the Leavitt path algebra can be connected via the Lie bracket algebra. Moreover, we use the Gelfand–Kirillov dimension for the Leavitt path algebra for a classification of nilpotency and solvability. [ABSTRACT FROM AUTHOR]

In this paper, we show that every finite-dimensional Zinbiel algebra over an arbitrary field is nilpotent, extending a previous result that they are solvable by other authors. [ABSTRACT FROM AUTHOR]

Tensor product of highest weight modules and intermediate modules for Virasoro algebra have been studied around 1997. Since then the irreducibility problem for tensor product of modules is open. We consider the loop-Virasoro algebra V i r ⊗ B , where Vir is the Virasoro algebra and B is a commutative associative unital algebra over ℂ. In this paper, we study the irreducibility problem for the tensor product of highest weight modules and intermediate modules for Vir ⊗ B. Finally we found out a necessary and sufficient condition for such modules to be isomorphic. [ABSTRACT FROM AUTHOR]

This paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra ℝ . This description implies that the space of all local derivations on ℝ when equipped with Lie bracket is isomorphic to the Lie algebra 7 (ℝ) of all real skew-symmetric 7 × 7 -matrices. We also consider 2 -local derivations on an octonion algebra over an algebraically closed field of characteristic zero and prove that every 2 -local derivation on is a derivation. Further, we apply these results to similar problems for the simple seven-dimensional Malcev algebra. As a corollary, we obtain that the real octonion algebra ℝ and Malcev algebra M 7 (ℝ) are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation. [ABSTRACT FROM AUTHOR]

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 along a -bimodule , J. Algebra Appl. 17(9) (2018) 1850169-1–1850169-21]. In this paper, under certain conditions, we show that the amalgamated Banach algebra ⋊ is BSE-algebra if and only if and are BSE-algebras. [ABSTRACT FROM AUTHOR]

This paper aims to present, in a unified manner, Cramer's rule which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies Cramer's rule for the system of v-quaternionic linear equations by means of a complex matrix representation of v-quaternion matrices, and gives an algebraic technique for solving the system of v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for Cramer's rule in quaternionic and split quaternionic mechanics. [ABSTRACT FROM AUTHOR]

Two presentations of quantum affine superalgebras were introduced by Yamane in [On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390], which were called Drinfeld–Jimbo realization and Drinfeld realization. Drinfeld realization contains infinite sequences of generators and relations. In this paper, we consider the Drinfeld realization of quantum affine superalgebra q ( ̂ m | n) associated to type m | n and define a simple algebra 0 ( ̂ m | n) generated by only a finite part of these sequences of quantum affine superalgebra q ( ̂ m | n). We show that the algebra 0 ( ̂ m | n) is isomorphic to the quantum affine superalgebra q ( ̂ m | n). Using the above isomorphism, we prove there exists an isomorphism between the two realizations. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of 2-nil ideal introduced by Yetkin Celikel in [E. Yetkin Celikel, 2-nil ideals of commutative rings, Bulletin of the Belgian Mathematical Society Simon Stevin 28 (3) (2021).] to 2-nil submodule which is a subclass of 2-absorbing primary submodules. Let M be an R -module and N be a proper submodule of M. We say that N is a 2-nil submodule of M if whenever a , b ∈ R , m ∈ M and a b m ∈ N , then a b ∈ Nil (M) or a m ∈ N or b m ∈ N. We study the properties of this concept and establish several characterizations. We also investigate the 2-nil ideals of amalgamation. The obtained results yield new original families of examples of 2-nil ideals. [ABSTRACT FROM AUTHOR]

Let α = (A g , α g) g ∈ be a group-type partial action of a connected groupoid on a ring A = ⊕ z ∈ 0 A z and B : = A ⋆ α be the corresponding partial skew groupoid ring. In the first part of this paper, we investigate the relation of several ring theoretic properties between A and B. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action λ , we characterize when λ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general partial skew groupoid rings. Several examples that illustrate the results on Leavitt path algebras are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider n -dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra. [ABSTRACT FROM AUTHOR]

A commutative ring R has the unique decomposition into ideals (UDI) property if, for any R -module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In [P. Goeters and B. Olberding, Unique decomposition into ideals for Noetherian domains, J. Pure Appl. Algebra 165 (2001) 169–182], the UDI property has been characterized for Noetherian integral domains. In this paper, we aim to study the UDI-like property for strong Mori domains; domains satisfying the ascending chain condition on w -ideals. [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]

Nottingham algebras are a class of just-infinite-dimensional, modular, ℕ -graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree 1 , and the second occurs in degree q , a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to ∞. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra L can be assigned a type, in such a way that the degrees and types of the diamonds completely describe L. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals q − 1. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type ∞. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Artin–Schelter regular algebras of dimension 5 with three generators in degree 1 under the hypothesis that GKdim ≥ 4 , in which the degree types of the relations for the number of the generating relations less than five can be determined. We prove that the only possible degree type of three generating relations is (2, 2, 3) and the only possible degree type of four generating relations is (2, 2, 3, 4). [ABSTRACT FROM AUTHOR]

In [S. Das, M. K. Sen and S. K. Maity, Leavitt path algebras for power graphs of finite groups, J. Algebra Appl., doi:10.1142/S0219498822502097], we studied several 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. In this paper, we compute the Grothendieck group of purely infinite simple Leavitt path algebra of the directed punctured power graph of a finite group. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a kind of double centralizer property for general linear supergroups. For the super space V = m | n over an algebraically closed field whose characteristic is not equal to 2 , we consider its ℤ 2 -homogeneous one-dimensional extension V ̲ = V ⊕ v , and the natural action of the supergroup G ̃ : = GL (V) × G m on V ̲. Then we have the tensor product supermodule ( V ̲ ⊗ r , ρ r ) of G ̃. We present a kind of generalized Schur–Sergeev duality which is said that the Schur superalgebras S ′ (m | , n , r) of G ̃ and the so-called weak degenerate double Hecke algebra ℋ ̲ r are double centralizers. The weak degenerate double Hecke algebra is an infinite-dimensional algebra, which has a natural representation on the tensor product space. This notion comes from [B. Shu, Y. Xue and Y. Yao, On enhanced reductive groups (I): Parabolic Schur algebras and the dualities related to degenerate double Hecke algebras, preprint (2013), arXiv:2005.13152 [Math. RT]], with a little modification. [ABSTRACT FROM AUTHOR]

In this paper, we introduced two classes of elements in the enveloping algebra U (g l (n)) : the double Young–Capelli bitableaux  [ S | T ] and the central Schur elements  S λ (n) , that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element S λ (n) is the sum of all double Young–Capelli bitableaux [ S | S ] , S row (strictly) increasing Young tableaux of shape λ ̃. The Schur elements S λ (n) are proved to be the preimages — with respect to the Harish-Chandra isomorphism — of the shifted Schur polynomials  s λ | n ∗ ∈ Λ ∗ (n). Hence, the Schur elements are the same as the Okounkov quantum immanants, recently described by the present authors as linear combinations of Capelli immanants. This new presentation of Schur elements/quantum immanants does not involve the irreducible characters of symmetric groups. The Capelli elements H k (n) are column Schur elements and the Nazarov elements I k (n) are row Schur elements. The duality in ζ (n) follows from a combinatorial description of the eigenvalues of the H k (n) on irreducible modules that is dual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the I k (n). The passage n → ∞ for the algebras ζ (n) is obtained both as direct and inverse limit in the category of filtered algebras, via the Olshanski decomposition/projection. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the structure of irreducible modules in the variety ℳ of commutative power-associative nilalgebras of nilindex ≤ 4. If A ∈ ℳ with dimension at most 5, then we prove that A 2 is contained in the annihilator of every irreducible A -module in the variety ℳ. Also, we consider the enveloping algebra of an algebra A in the variety ℳ and we obtain a new example of a commutative power-associative non-nilpotent nilalgebra of dimension 9. [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]

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]

Let F be a field of char ≠ 2 with finitely many square classes. In this paper, we define a new rational valued invariant of F , and call it the division probability of F. We compute it for all fields of elementary type. Further, we show that n (q) ≤ e (q) , where e (q) is the number of Witt-equivalence classes of fields with | F × / F × 2 | = q , and n (q) is the count of rational numbers that appear as division probabilities for fields F of elementary type with | F × / F × 2 | = q. In the paper, we also determine n (q) for all q ≤ 2 1 6 and show that rational numbers of type (2 r − 1) / 2 r + 1 always occur as division probability for a suitable field F. [ABSTRACT FROM AUTHOR]

In this paper, we extend the result proved by Ulbrich about the characterization of Galois extensions linked to group algebras upon the non-associative (quasigroup) quasigroupoid magma setting. Also, as a particular instance of the results contained in this paper, we obtain the ones proved for Galois extensions related with groupoid algebras. [ABSTRACT FROM AUTHOR]

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 + / (I 2 (t 2 + a t + 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. [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 re-establish a bialgebraic theory on a Hom–Leibniz algebra by a new dual representation, which weakens the involutive condition on the structure map in [S. Guo, S. X. Wang and X. H. Zhang, The Classical Hom–Leibniz Yang–Baxter Equation and Hom–Leibniz Bialgebras, Mathematics 10 (2022) 1920; I. Laraiedh and S. Silvestrov, Hom–Leibniz bialgebras and BiHom–Leibniz dendriform algebras, Afr. Mat. 34 (2023), Article ID: 28, 45 pp]. We also give the classification of triangular Hom–Leibniz bialgebras on two-dimensional Hom–Leibniz algebras induced by “Yau twist principle”. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra 픰픠픥n := 픰픩2 ⋉ 픥n is the semidirect product of the Lie algebra 픰픩2 with the n-th Heisenberg Lie algebra 픥n. In this paper, we will show that 픰픠픥n has neither nontrivial 1 2-derivations nor nontrivial transposed Poisson algebra structures, and doesn’t have nonzero 1 2-biderivations. In addition, all δ-derivations and δ-biderivations on 픰픠픥n are all described. [ABSTRACT FROM AUTHOR]