Journal of Algebra and Its Applications

Let $G$ be a group and let $S={\oplus }_{g\in G}{S}_g$ be a $G$-graded ring. Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading of $S$. It is well known that if $S$ is strongly $G$-graded, then the induced $G/N$-grading is strong for any $N$. The class of epsilon-strongly graded rings was recently introduced by Nystedt et al. as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and sufficient conditions for the induced $G/N$-grading of an epsilon-strongly $G$-graded ring to be epsilon-strong. Our method involves relating different types of rings equipped with local units ($s$-unital rings, rings with sets of local units, rings with enough idempotents) with generalized epsilon-strongly graded rings.

Let $p$ be a prime and $G$ a finite $p$-group. Hall proved ${\delta }^k(G)\equiv 0(mod{p}^{d(G)-k+1})$ for $0\le k\le d(G)$, where ${\delta }^k(G)$ is the number of subgroups of index $p^k$ which do not contain the Frattini subgroup of $G$. We determine the minimal values of ${\delta }^k(G)$ for $p>2$ and ${\delta }^k(G)>0$.

In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph homomorphisms, and obtain a covariant functor from a graph category to an algebra category.

Given a directed graph $E$ and an associative unital ring $R$ one may define the Leavitt path algebra with coefficients in $R$, denoted by $L_R(E)$. For an arbitrary group $G$, $L_R(E)$ can be viewed as a $G$-graded ring. In this paper, we show that $L_R(E)$ is always nearly epsilon-strongly $G$-graded. We also show that if $E$ is finite, then $L_R(E)$ is epsilon-strongly $G$-graded. We present a new proof of Hazrat’s characterization of strongly $\mathbb{Z}$-graded Leavitt path algebras, when $E$ is finite. Moreover, if $E$ is row-finite and has no source, then we show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ has no sink. We also use a result concerning Frobenius epsilon-strongly $G$-graded rings, where $G$ is finite, to obtain criteria which ensure that $L_R(E)$ is Frobenius over its identity component.

In this paper, by using linear translators, we characterize a new class of permutation polynomials of the form $x+{\sum }_{i=1}^k{\gamma }_i{h}_i(f_i(x))$, which has a more general form than $x+\gamma h(f(x))$. Then, we present the compositional inverses of such permutation polynomials. Furthermore, by specifying the functions $h_i(x)$ and $f_i(x)$, we can get some new permutation polynomials of the forms $x+{\gamma }_1{(Tr({\beta }_{1}x)+{\delta }_1)}^{s_1}+{\gamma }_2{(Tr({\beta }_{2}x)+{\delta }_2)}^{s_2}$ and $x+{\gamma }_1{(Tr(x)+\delta )}^{s_1}+{\gamma }_2{(Tr(x)+\delta )}^{s_2},$ where $Tr(x)$ is the trace function.

Let $G$ be a finite $p$-group and let ${\text{Aut}}_c(G)$ be the set of all central automorphisms of $G.$ For any group $G$, the center of the inner automorphism group, $Z(\text{Inn}(G))$, is always contained in ${\text{Aut}}_c(G).$ In this paper, we study finite $p$-groups $G$ for which ${\text{Aut}}_c(G)$ is of minimal possible, that is ${\text{Aut}}_c(G)=Z(\text{Inn}(G)).$ We characterize the groups in some special cases, including $p$-groups $G$ with $C_G(Z(\mathrm{\Phi }(G)))\ne \mathrm{\Phi }(G)$, $p$-groups with an abelian maximal subgroup, metacyclic $p$-groups with $p>2$, $p$-groups of order $p^m$ and exponent $p^{m-2}$ and Camina $p$-groups.

We exhibit the first example of a non-rigid complex filiform Lie algebra with all its codimension 1 ideals being characteristically nilpotent by constructing an explicit nontrivial filiform deformation of it.

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ($\mathrm{\text{ADE}}$) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.

In analogy to the elementwise definitions of UU- and UJ-rings, a ring $R$ is called UNJ if $1+N(R)+J(R)=U(R)$. After presenting several characterizations and properties, we consider the UNJ-rings within many well-studied classes of rings. In particular, we examine Dedekind finite rings, 2-primal rings, (semi)regular rings, $\pi$-regular rings and rings that satisfy the identity $x^2=x$. Finally, we conclude this paper with group UNJ-rings.

Let $R\subseteq S$ be an extension of integral domains with only finitely many intermediate rings, where $R$ is not a field and $S$ is not necessarily the quotient field of $R$ or $R$ is not necessarily integrally closed in $S$. In this paper, we exactly determine the number of intermediate rings between $R$ and $S$ and give a way to compute it.

Generalizing the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC), a class of constraints involving a pair of operators was considered in [Extensions of operators, liftings of monads, and mixed distributive laws, Appl. Categ. Struct. 26 (2018) 747–765]. For a given constraint, the existences of extensions of differential and Rota–Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are shown to be equivalent. In this paper, we give a classification of the constraints satisfying these equivalent conditions.

Let ${\mathcal{𝒢}}_R$ denote the unitary homogeneous bi-Cayley graph over a finite commutative ring $R$. In this paper, we determine the energy of ${\mathcal{𝒢}}_R$ and that of its complement and line graph, and characterize when such graphs are hyperenergetic. We also give a necessary and sufficient condition for ${\mathcal{𝒢}}_R$ (respectively, the complement of ${\mathcal{𝒢}}_R$, the line graph of ${\mathcal{𝒢}}_R$) to be Ramanujan.

Let $X$ and $Y$ be $N$-complexes with $N\ge 2$ an integer such that $X$ has finite Gorenstein projective dimension and $Y$ has finite Gorenstein injective dimension. We define the $n$th Gorenstein cohomology groups

We prove a conjecture proposed in [A. De Sole, V. G. Kac and D. Valeri, Finite $W$-algebras for ${𝔤𝔩}_N$, Adv. Math. 327 (2018) 173–224.] describing the Lax type operator $L(z)$ for the quantum finite $W$-algebras of ${𝔤𝔩}_N$ in terms of a PBW generating system for the $W$-algebra. In doing so, we extend this result to an arbitrary good grading and an arbitrary isotropic subspace of $𝔤[\frac{1}{2}]$.

In this paper, the categories of braided categorical associative algebras and braided crossed modules of associative algebras are studied. These structures are also correlated with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras.

Introducing $h$- and $h^{\prime }$-deformations of ${\mathbb{Z}}_2$-graded ($1+2$)- and ($2+1$)-spaces, denoted by ${𝔸}_h^{1|2}$ and ${𝔸}_{h^{\prime }}^{2|1}$, a two-parameter differential calculus, the quantum de Rham complex, on ${𝔸}_h^{1|2}$ are explicitly constructed. It is shown that in contrast to the standard $q$-deformation of ${𝔸}^{1|2}$, the above complex is unique for ${𝔸}_h^{1|2}$. The quantum de Rham complex of ${𝔸}_h^{1|2}$ is discussed via a contraction using standard one.

In this paper, we define the direct sum relation on a module and prove that this is a partial order when the module is regular. We also focus on a regular support and a strong regular support of an element in a module, which are extensions of a generalized inverse and a reflexive generalized inverse of an element in a ring, respectively. Using regular supports and strong regular supports, we present various characterizations of the direct sum order as an application. We obtain some decompositions of a module via the direct sum order.

It is proved that if a left brace $A$ has the operation ∗ associative, then $A$ is a two-sided brace. Consequently, $A$ is a Jacobson radical ring. This answers a question of Cedó, Gateva-Ivanova and Smoktunowicz.

An $F$-algebra $A$ with unit $1$ is said to be a locally matrix algebra if an arbitrary finite collection of elements $a_1,\dots ,a_s$ from $A$ lies in a subalgebra $B$ with $1$ of the algebra $A$, that is isomorphic to a matrix algebra $M_n(F)$, $n\ge 1$. To an arbitrary unital locally matrix algebra $A$, we assign a Steinitz number $st(A)$ and study a relationship between $st(A)$ and $A$.

Denote by $S$ a polynomial ring over a field, and let $M=(m_1,\dots ,m_q)$ be a monomial ideal of $S$. If $\mathrm{\text{codim}}(S/M)=1$, we prove that the multiplicity of $S/M$ is given by