Narrow Results

This paper is devoted to the study of local and 2-local derivations of null-filiform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation.

In this paper, we first introduce a weighted derivation on algebras over an operad ${𝒫}$, and prove that for the free ${𝒫}$-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ($q$-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential ($q$-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.

In this paper, we study the derivations of group algebras of some important groups, namely, Dihedral ($D_{2n}$), Dicyclic ($T_{4n}$) and Semi-dihedral ($S{D}_{8n}$). First, we explicitly classify all inner derivations of a group algebra $𝔽G$ of a finite group $G$ over an arbitrary field $𝔽$. Then we classify all $𝔽$-derivations of the group algebras $𝔽D_{2n}$, $𝔽T_{4n}$ and $𝔽(S{D}_{8n})$ when $𝔽$ is a field of characteristic $0$ or an odd rational prime $p$ by giving the dimension and an explicit basis of these derivation algebras. We explicitly describe all inner derivations of these group algebras over an arbitrary field. Finally, we classify all derivations of the above group algebras when $𝔽$ is an algebraic extension of a prime field.

For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal $L$ of a ring $R$, we offer a criterion for $R$ to be $L$-semiprime. For a prime ring $R$, we characterizes Lie ideals $L$ of $R$ such that $R$ is $L$-semiprime. Moreover, $X$-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied.

In this paper, we study the generators of the module of derivations of the ring of invariants for some dihedral groups. Let $k$ be an algebraically closed field of characteristic $0$ and $D_{n,q}(\subset \mathrm{\text{GL}}(2,k))$ be a finite dihedral group. Let $R=k{[X,Y]}^{D_{n,q}}$ be the ring of invariants obtained by the linear action of $D_{n,q}$ on $k[X,Y]$. In [ R. V. Gurjar and A. Patra, On minimum number of generators for some derivation modules, J. Pure Appl. Algebra$226$(11) (2022) 107105], Gurjar–Patra proved that $\mu ({\text{Der}}_k(R))<\left|D_{n,q}\right|$. We will give a better bound on $\mu ({\text{Der}}_k(R))$ in this paper.

The main goal of this paper is to extend the main result of [22] to non-unital triangular rings. Indeed, we prove that every Jordan higher derivation on a triangular ring is a higher derivation without requiring the existence of unity. As a remark, the modification of [12, Theorem 3] is also discussed.

This paper investigates the algebraic properties of the Bondi–Metzner–Sachs (BMS) superalgebra, as initially introduced by G. Barnich, L. Donnay, J. Matulich and R. Troncoso, Asymptotic symmetries and dynamics of 3D flat supergravity, J. High Energy Phys.2014 (2014) 071. We introduce a new realization of the BMS superalgebra using the Balinsky–Novikov construction. Utilizing this approach, we prove that the BMS superalgebra constitutes the unique minimal supersymmetric extension of the BMS algebra in three dimensions. Additionally, we compute the low-order cohomology groups of the BMS superalgebra and classify its derivations, central extensions, and automorphisms.

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions , where , d_l = 2^il + 2^jl + 1, m, i_l and j_l are positive integers, n > i_l > j_l. Furthermore, for a class of Boolean functions we deduce a tighter lower bound on the second order nonlinearity of the functions, where , d_l = 2^i_lγ + 2^j_lγ + 1, i_l > j_l and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1.

Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by f_μ(x) = Tr(μx^2i+2j+1), , i and j are positive integers such that i > j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions f_μ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.

In this paper, we study the structure theory of two classes of Lie superalgebras ${𝒮}(q)$ of Block type, where $q$ is a positive integer. In particular, the automorphism groups, derivation superalgebras and central extensions of ${𝒮}(q)$ are completely determined.

This paper is devoted to studying 2-local derivations on the planar Galilean conformal algebra. We prove that every 2-local derivation on the planar Galilean conformal algebra is a derivation.

We study derivations on a smooth manifold, its twisted de Rham cohomology, generalized connections on vector bundles and their characteristic classes.

This note proposes a non-inertial similarity solution for the classic von Kármán swirling flow as perceived from the rotational frame. The solution is obtained by implementing non-inertial similarity parameters in the non-inertial boundary layer equations. This reduces the partial differential equations to a set of ordinary differential equations that is solved through an integration routine and shooting method.

In this paper we describe a share package of functions for computing with finite, permutation crossed modules, cat1-groups and their morphisms, written using the group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat1-groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module , and the corresponding monoid of sections of a cat1-group , are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of X, are used to construct the actor crossed module of X which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat1-structures on groups of order at most 30 in [2].

The study of prime ideals has been an area of active research. In recent past a considerable work has been done in this direction. Associated prime ideals and minimal prime ideals of certain types of Ore extensions have been characterized.

In this paper a relation between completely prime ideals of a ring R and those of R[x; σ, δ] has been given; σ is an automorphisms of R and δ is a σ-derivation of R. It has been proved that if P is a completely prime ideal of R such that σ(P) = P and δ(P) ⊆ P, then P[x; σ, δ] is a completely prime ideal of R[x; σ, δ]. It has also been proved that this type of relation does not hold for strongly prime ideals.

We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

We show that any local derivation on the solvable Leibniz algebras with model or abelian nilradicals, whose dimension of complementary space is maximal is a derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have $1$ dimension complementary space, admit local derivations which are not derivations. Moreover, similar problem concerning $2$-local derivations of such algebras is investigated and an example of solvable Leibniz algebra is given such that any $2$-local derivation on it is a derivation, but which admits local derivations which are not derivations.

Let $P$ and $Q$ be finite posets and $R$ a commutative unital ring. In the case where $R$ is indecomposable, we prove that the $R$-linear isomorphisms between partial flag incidence algebras $I^3(P,R)$ and $I^3(Q,R)$ are exactly those induced by poset isomorphisms between $P$ and $Q$. We also show that the $R$-linear derivations of $I^3(P,R)$ are trivial.

This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.

The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions.

We prove that every derivation and every locally nilpotent derivation of the subalgebra $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$, where $n\ge 2$, of the polynomial algebra $K[x,y]$ in two variables over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Moreover, we prove that every automorphism of $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$ over an algebraically closed field $K$ of characteristic zero is induced by an automorphism of $K[x,y]$. We also show that the group of automorphisms of $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$ admits an amalgamated free product structure.