Algebra Colloquium

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, ${\Omega }_1(P\cap G^{\prime })\le Z(P)$ and N_G(P) is 2-nilpotent. In this paper, it is shown that SL₂(q), $q>3$, is a special local 2-nilpotent group if and only if $q^2\equiv 1\left(\mathrm{mod}16\right)$, and that GL₂(q), $q>3$, is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

An algebra is called skew-symmetric if its multiplication operation is a skewsymmetric bilinear application. We determine all these algebras in dimension 3 over a field of characteristic different from 2. As an application, we determine the subvariety of 3-dimensional Hom-Lie algebras. For this type of algebra, we study also the case of dimension 4.

Herzog, Hibi, Hreindóttir et al. introduced the class of closed graphs, and they proved that the binomial edge ideal J_G of a graph G has quadratic Gröbner bases if G is closed. In this paper, we introduce the class of weakly closed graphs as a generalization of the closed graph, and we prove that the quotient ring S/J_G of the polynomial ring $S=K[x_1,\dots ,x_n,y_1,\dots ,y_n]$ with K a field and $n=|V(G)|$ is F-pure if G is weakly closed. This fact is a generalization of Ohtani’s theorem.

For a positive integer n, we denote by π(n) the set of all prime divisors of n. For a finite group G, the set $\pi (G):=\pi (|G|)$ is called the prime spectrum of G. Let $M⋖G$ mean that M is a maximal subgroup of G. We put $K(G)=\max \left\{\right|\pi (G)\backslash \pi (M)|:M⋖G\}$ and $k(G)=\min \left\{\right|\pi (G)\backslash \pi (M)|:M⋖G\}$. In this notice, using well-known number-theoretical results, we present a number of examples to show that both K(G) and k(G) are unbounded in general. This implies that the problem “Are k(G) and K(G) bounded by some constant k?”, raised by Monakhov and Skiba in 2016, is solved in the negative.

Generalizing the notion of strongly nil clean rings, we introduce strongly quasi-nil clean rings. Some fundamental properties and equivalent characterizations of this class of rings are provided. By means of g-Drazin inverses, Cline’s formula and Jacobson’s lemma for strongly quasi-nil clean elements are investigated.

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz), is introduced and studied. It is shown that the $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring $R[[x^{-1};\delta ]]$ to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

In this paper, we introduce the notion of excellent extensions of rings. Let Γ be an excellent extension of an Artin algebra Λ, we prove that Λ satisfies the Gorenstein symmetry conjecture (resp., finitistic dimension conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture) if and only if so does Γ. As a special case of excellent extensions, when G is a finite group whose order is invertible in Λ acting on Λ and Λ is G-stable, we prove that if the skew group algebra ΛG satisfies strong Nakayama conjecture (resp., generalized Nakayama conjecture), then so does Λ.

In this paper, the structures of dual Lie bialgebras of twisted Schrödinger-Virasoro type are investigated. By studying the maximal good subspaces, we determine the dual Lie coalgebras of the twisted Schrödinger-Virasoro algebras. Then based on this, we construct the dual Lie bialgebra structures of this type. As by-products, four new infinite dimensional Lie algebras are obtained.

In this short note, we study the rank of a restricted Lie algebra (𝔤, [p]), and give some applications which concern the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra.

In this paper, the containment problem for the defining ideal of a special type of zero-dimensional subscheme of ℙ², the so-called quasi star configuration, is investigated. Some sharp bounds for the resurgence of these types of ideals are given. As an application of this result, for every real number $0<\epsilon <\frac{1}{2}$, we construct an infinite family of homogeneous radical ideals of points in 𝕂[ℙ²] such that their resurgences lie in the interval [2−ε, 2). Moreover, the Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of quasi star configurations are determined. In particular, it is shown that all of these ordinary powers have a linear resolution.

Let G be a finite group and $\mathcal{H}$ be a Hartley set of G. In this paper, we prove the existence and conjugacy of $\mathcal{H}$-injectors of G and describe the characterization of injectors via radicals. As applications, some known results are directly followed.

Let R be a prime ring of characteristic different from 2, Q be its maximal right ring of quotients, and C be its extended centroid. Suppose that $f(x_1,\dots ,x_n)$ is a non-central multilinear polynomial over C, $0\ne p\in R$, and F, G are two b-generalized derivations of R. In this paper we describe all possible forms of F and G in the case $pG(F(f(r))f(r))=0$ for all $r=(r_1,\dots ,r_n)$ in Rⁿ.

If H is a subgroup of a finite group G, then the automizer Aut_G(H) of H in G is defined as the group of automorphisms of H induced by conjugation by elements of N_G(H). A finite group G is called an NNC-group if for any non-normal abelian subgroup A, either $Au{t}_G(A)=1$ or $C_G(A)=A$. In this paper, classifications of nilpotent NNC-groups and non-solvable NNC-groups are given. We also investigate the solvable NNC-groups and describe the structure of solvable NNC-groups.

In this paper, by using the Anick resolution and Gröbner-Shirshov basis for quantized enveloping algebra of type B₂, we compute the minimal projective resolution of the trivial module of $U_q^{+}(B_2)$, and as an application we compute the global dimension of $U_q^{+}(B_2)$.