Journal of Algebra and Its Applications

In this paper we discuss some properties of abelian (weakly) nil clean rings. We prove that any subring of an abelian (weakly) nil clean ring is (weakly) nil clean (Theorem 2). We also show that the tensor product of commutative (weakly) nil clean rings is also (weakly) nil clean and give sufficient conditions for the converse to be true (Theorems 3–6).

We focus on a class of filtered quantum algebras ${𝔅}_q(𝔤)$ which are both coideal subalgebras of quantum groups and Poincaré–Birkhoff–Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222–249], Xu and Yang proved that braid group actions on ${𝔅}_q(𝔤)$ introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for ${𝔅}_q(𝔤)$. In this present paper, for each element $w$ in the Weyl group of $𝔤$ we first introduce a subspace ${𝔅}^w$ and a subalgebra $𝔅(w)$ of ${𝔅}_q(𝔤)$, where $𝔅(w)$ can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on $w$ is given for ${𝔅}^w=𝔅(w)$. Moreover, we prove that $𝔅(w_1)=𝔅(w_2)$ if and only if $w_1$ and $w_2$ can be generated by the same simple reflections. Finally, we characterize the algebra $𝔅(w)$ which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

Let $R$ be a left and right Noetherian ring. In this paper, we prove that any Gorenstein transpose of a finitely generated $R$-module is exactly an Auslander transpose. As applications, we obtain a new relation between a Gorenstein transpose of a module with a transpose of the same module, and show that the Gorenstein transpose of a module is unique up to Gorenstein projective equivalence. In addition, when $R$ is an Artin algebra, the corresponding Auslander–Reiten sequences are constructed in terms of Gorenstein transposes.

In this paper, we continue to study the differential inverse power series ring $R[[x^{-1};\delta ]]$, where $R$ is a ring equipped with a derivation $\delta$. We characterize when $R[[x^{-1};\delta ]]$ is a local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, right stable range one, abelian, projective-free, $I$-ring, respectively. Furthermore, we prove that $R[[x^{-1};\delta ]]$ is a domain satisfying the $ACC$ on principal left ideals if and only if so does $R$. Also, for a piecewise prime ring (PWP) $R$ we determine a large class of the differential inverse power series ring $R[[x^{-1};\delta ]]$ which have a generalized triangular matrix representation for which the diagonal rings are prime. In particular, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does $R[[x^{-1};\delta ]]$. Our results extend and unify many existing results.

A nonzero ring is called a UN-ring if every nonunit is a product of a unit and a nilpotent element. We show that all simple Artinian rings are UN-rings and that the UN-rings whose identity is a sum of two units (e.g. if 2 is a unit), form a proper class of 2-good rings (in the sense of P. Vámos). Thus, any noninvertible matrix over a division ring is the product of an invertible matrix and a nilpotent matrix.

Glauberman and Wright in [G. G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra 8 (1968) 415–417] proved that a nilpotent Moufang loop is the direct product of $p$-loops for some primes $p$, consequently the elements of coprime order commute in a nilpotent Moufang loop. In this paper, we prove that in Moufang loops of odd order this condition is equivalent to the central nilpotence.

Let $G$ be a finite group and $I(G)$ be the set of the elements $x$ of $G$ such that $x^2=1$. In this paper, we characterize all groups $G$ such that $|I(G)|\ge \frac{\left|G\right|}{3}$ and $G$ is not a 2-group.

In this paper we describe a package XModAlg for GAP4 which constructs crossed modules of k-algebras and cat¹-algebras over k, and their morphisms. We include a table of cat¹-algebras.

Let $\mathrm{\text{cd}}(G)$ be the set of all irreducible complex characters of a finite group $G$. In [K. Aziziheris, Determining group structure from sets of irreducible character degrees, J. Algebra 323 (2010) 1765–1782], we proved that if $m$ and $n$ are relatively prime integers greater than $1$, $p$ is prime not dividing $mn$, and $G$ is a solvable group such that $\mathrm{\text{cd}}(G)=\{1,p,n,m,pn,pm\}$, then under some conditions on $p,m,$ and $n$, the group $G=A\times B$ is the direct product of two normal subgroups, where $\mathrm{\text{cd}}(A)=\{1,p\}$ and $\mathrm{\text{cd}}(B)=\{1,n,m\}$. In this paper, we replace $p$ by $u$, where $u>1$ is an arbitrary positive integer, and we obtain similar result. As an application, we show that if $G$ is a finite group with $\mathrm{\text{cd}}(G)=\{1,21,13,55,273,1155\}$ or $\mathrm{\text{cd}}(G)=\{1,15,391,58,5865,870\}$, then $G$ is a direct product of two non-abelian normal subgroups.

We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and discuss some open questions. In particular, we provide shorter and clearer proofs of the main results of this theory. We also discuss some new theoretical and computational results and conjectures on the lower central series of the free algebra in two generators modulo a generic homogeneous relation.

In the mid 1970s, Michel Racine classified the maximal subalgebras of an octonian algebra. In this paper, we classify the maximal associative subalgebras. It turns out that there are four, up to isomorphism, all of dimension $4$. In final sections, we apply our findings to investigate the groups that sit inside the Moufang loop of invertible elements of the split octonians and also to show that a well-known inequality of Jørgensen holds in a new context.

Let $N$ be a nontrivial $2$-closed group and let $H$ be an arbitrary permutation group on a finite set $\mathrm{\Omega }$. Let $G=N\wr H$ be the corresponding permutational wreath product of $N$ by $H$. It is shown that every class-preserving Coleman automorphism of $2$-power order of $G$ is inner. As a direct consequence, it is obtained that the normalizer property holds for $G$. Further, it is shown that every class-preserving Coleman automorphism of $G$ is inner whenever $N$ is nilpotent. Our results generalize some known ones.

All solvable, indecomposable, finite-dimensional, complex Lie superalgebras $𝔤$ whose first derived ideal lies in its nilradical, and whose nilradical is a Heisenberg Lie superalgebra $𝔥$ associated to a ${\mathbb{Z}}_2$-homogeneous supersymplectic complex vector superspace $V$, are here classified up to isomorphism. It is shown that they are all of the form $𝔤=𝔥\oplus 𝔞$, where $𝔞$ is even and consists of non-${ad}_{𝔤}$-nilpotent elements. All these Lie superalgebras depend on an element $\gamma$ in the dual space ${𝔞}^{\ast }$ and on a pair of linear maps defined on $𝔞$, and taking values in the Lie algebras naturally associated to the even and odd subspaces of $V$; namely, if the supersymplectic form is even, the pair of linear maps defined on $𝔞$ take values in $𝔰𝔭(V_0)$, and $𝔬(V_1)$, respectively, whereas if the supersymplectic form is odd these linear maps take values on $𝔤𝔩(V_0)\simeq 𝔤𝔩(V_1)$. When the supersymplectic form is even, a bilinear, skew-symmetric form defined on $𝔞$ is further needed. Conditions on these building data are given and the isomorphism classes of the resulting Lie superalgebras are described in terms of appropriate group actions.

Given a Lie algebra $𝔤$, let $\mu (𝔤)$ and ${\mu }_{\mathrm{\text{nil}}}(𝔤)$ be the minimal dimension of a faithful representation and nilrepresentation of $𝔤$, respectively. In this paper, we give $\mu (𝔤)$ and ${\mu }_{\mathrm{\text{nil}}}(𝔤)$ for each nilpotent Lie algebra $𝔤$ of dimension $6$ over a field $𝕂$ of characteristic zero. We also give a minimal faithful representation and nilrepresentation in each case.

We describe two methods to determine all generators of the additive semigroup of the non-negative exponent $n\times n$-matrices, and illustrate them finding all generating $3\times 3$-exponent matrices. The generating $4\times 4$-exponent matrices are found using a computer. We consider the Hasse diagram $H$ of the partially ordered set of non-negative matrices and prove that for an arbitrary non-negative exponent matrix $A$ there exists an oriented path in $H,$ starting in some matrix unit and ending in $A,$ which does not pass through any other exponent matrix. We also show that for any non-negative exponent matrix $A$ there exists a chain of non-negative exponent matrices $A_0\le A_1\le \cdots \le A_t=A$ such that $A_0$ is a $(0,1)$-matrix, and each $A_i$ is obtained from $A_{i-1}$ by adding a $(0,1)$-matrix.

Let ${}_S{C}_R$ be a semidualizing bimodule with $S$ left coherent and $R$ right coherent. For a non-negative integer $n$, it is shown that $l.FP$-${\mathrm{\text{id}}}_S(C)=r.FP$-${\mathrm{\text{id}}}_R(C)\le n$ if and only if every finitely presented left $S$-module has $G_C$-projective dimension at most $n$ if and only if every finitely presented right $R$-module has $G_C$-projective dimension at most $n$. As applications, some well-known results are extended.

By using the language of cogroupoids, we show that Hopf–Galois objects of a twisted Calabi–Yau (CY) Hopf algebra with bijective antipode are still twisted CY, and give their Nakayama automorphism explicitly. As applications, cleft Galois objects of twisted CY Hopf algebras and Hopf–Galois objects of the quantum automorphism groups of non-degenerate bilinear forms are proved to be twisted CY.

In this paper, we show that total integrals and cointegrals are new sources of stable anti-Yetter–Drinfeld modules. We explicitly show that how special types of total (co)integrals can be used to provide both (stable) anti Yetter–Drinfeld and Yetter–Drinfeld modules. We use these modules to classify total (co)integrals and (cleft) Hopf Galois (co)extensions of the Connes–Moscovici Hopf algebra, and some examples of universal enveloping algebra and polynomial algebra.

Let $k$ be an arbitrary field of characteristic $0$. We prove that the group of automorphisms of a free Poisson field $P(x,y)$ in two variables $x,y$ over $k$ is isomorphic to the Cremona group ${\mathrm{\text{Cr}}}_2(k)$. We also prove that the universal enveloping algebra $P{(x_1,\dots ,x_n)}^e$ of a free Poisson field $P(x_1,\dots ,x_n)$ is a free ideal ring and give a characterization of the Poisson dependence of two elements of $P(x_1,\dots ,x_n)$ via universal derivatives.

For a finite group $G$, let $\omega (G)$ be the maximum size of a set of pairwise noncommuting elements in $G$. In this paper, we give an upper bound of $\omega (G)$ for an arbitrary nilpotent group $G$. As an application of this result, we give a partial answer to Question 2.8 of [A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000) 139–146]. Also we compute $\omega (G)$ when $\frac{G}{Z(G)}$ is a Frobenius group. Finally we describe structural properties of all groups $G$ with $\omega (G)=5$.