Journal of Algebra and Its Applications

By modifying the existing definition of a compressed zero-divisor graph ${\mathrm{\Gamma }}_E(K)$, we define a compressed zero-divisor graph $\mathrm{\Theta }(K)$ of a finite commutative unital ring $K$, where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of ${\mathrm{\Gamma }}_E(K)$). We prove that this is the best possible compression which induces a functor $\mathrm{\Theta }$, and that this functor preserves categorial products (in both directions). We use the structure of $\mathrm{\Theta }(K)$ to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

Related to a Coxeter group are certain sets of tangents of the deltoid with evenly distributed orientations forming simplicial line configurations. These configurations are used to construct curves and surfaces with $ADE$ singularities. Other surfaces associated with invariants of exceptional complex reflection groups are considered. A new lower bound for the maximal number of $D_4$ singularities in a sextic surface is obtained. Several Calabi–Yau threefolds defined as double coverings of the complex projective 3-space branched along nodal octic surfaces and Calabi–Yau quintic threefolds are analyzed. The Hodge numbers of a small resolution of all the nodes of the singular threefolds are obtained.

Given a non-commutative finite-dimensional $F$-central division ring $D$, $N$ a subnormal subgroup of ${\mathrm{\text{GL}}}_n(D)$ and $M$ a non-abelian maximal subgroup of $N$, then either $M$ contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup $A$ of $M$ such that $K=F[A]$ is a subfield of $M_n(D)$, $K/F$ is Galois and $\mathrm{\text{Gal}}(K/F)\cong M/(K^{\ast }\cap N)$, also $\mathrm{\text{Gal}}(K/F)$ is a finite simple group with $F[M]=M_n(D)$.

In this paper, we introduce a generalized inverse of a matrix, namely, the generalized WG inverse, which a generalization of the WG inverse. Several necessary and sufficient conditions such that a matrix to be generalized WG invertible are obtained. Moreover, the formulae of generalized WG inverse of a matrix are given. Finally, we give a algebraic form of the generalized WG inverse.

In this paper, we will study the notion of a Frobenius ⋆-algebra and prove some orthogonality relations for the irreducible characters of a Frobenius ⋆-algebra. Then we will study $\mathbb{Z}/m\mathbb{Z}$-graded Frobenius ⋆-algebras and prove some twisted orthogonality relations for them.

Let $p$ be a multilinear polynomial in several noncommuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra.

Let $R(a,b)$ be a Lie conformal algebra related to Galilean conformal algebras, where $a,b$ are complex numbers. All the conformal derivations of $R(a,b)$ are shown to be inner. The rank one conformal modules and $\mathbb{Z}$-graded free intermediate series modules over $R(a,b)$ are completely classified. The corresponding results of the finite conformal subalgebra of $R(a,b)$ are also obtained as byproducts.

A finite group $G$ is said to be a $T_{2,k}$-group for some integer $k\ge 1$, if there exist two distinct nontrivial degrees $a,b\in \mathrm{\text{cd}}(G)\setminus \{1\}$ such that the multiplicities of $a$ and $b$ are $2$ and $k$, respectively, and for every $d\in \mathrm{\text{cd}}(G)\setminus \{1,a,b\}$ the multiplicity of $d$ in $G$ is trivial. In this paper, we prove that if $G$ is a nonsolvable $T_{2,k}$-group, then $k\le 4$ and we determine all nonsolvable $T_{2,k}$-groups.

We prove that the zero-divisor graph of a direct product of matrices over finite zero-divisor free semirings uniquely determines the sizes of matrices and cardinalities of semirings in question. We also give an example that the semirings themselves are not necessarily uniquely determined.

In this paper, using the polynomial composition methods some computationally simple and explicit ways for constructing higher degrees primitive polynomials from a given primitive polynomial over ${𝔽}_q$ are given.

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of ${\mathbb{N}}^d$ with finite complement in ${\mathbb{N}}^d$. These semigroups are affine semigroups, which in particular implies that they are finitely generated. For a given finite set of elements in ${\mathbb{N}}^d$ we show how to deduce if the monoid spanned by this set is a generalized numerical semigroup and, if so, we calculate its set of gaps. Also, given a finite set of elements in ${\mathbb{N}}^d$ we can determine if it is the set of gaps of a generalized numerical semigroup and, if so, compute the minimal generators of this monoid. We provide a new algorithm to compute the set of all generalized numerical semigroups with a prescribed genus (the cardinality of their sets of gaps). Its implementation allowed us to compute (for various dimensions) the number of numerical semigroups of higher genus than has previously been computed.

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.

Let $R$ be a commutative Noetherian ring, $𝔞$ an ideal of $R$ and $M$ an $R$-module with $dimM=d$. We get equivalent conditions for top local cohomology module ${\mathrm{\text{H}}}_{𝔞}^d(M)$ to be Artinian and $𝔞$-cofinite Artinian separately. In addition, we prove that if $(R,𝔪)$ is a local ring such that ${\mathrm{\text{Ext}}}_R^i(R/𝔞,M)$ is minimax, for each $i\le d$, then ${\mathrm{\text{Ext}}}_R^i(N,M)$ is minimax $R$-module for each $i\ge 0$ and for each finitely generated $R$-module $N$ with $\mathrm{\text{dim}}N\le 2$ and ${\mathrm{\text{Supp}}}_R(N)\subseteq V(𝔞)$. As a consequence we prove that if $\mathrm{\text{dim}}R/𝔞=2$ and ${\mathrm{\text{Supp}}}_R(M)\subseteq \mathrm{\text{V}}(𝔞)$, then $M$ is $𝔞$-cominimax if (and only if) ${\mathrm{\text{Hom}}}_R(R/𝔞,M)$, ${\mathrm{\text{Ext}}}_R^1(R/𝔞,M)$ and ${\mathrm{\text{Ext}}}_R^2(R/𝔞,M)$ are minimax. We also prove that if $dimR/𝔞=2$ and $n\in {\mathbb{N}}_0$ such that ${\mathrm{\text{Ext}}}_R^i(R/𝔞,M)$ is minimax for all $i\le n+1$, then ${\mathrm{\text{H}}}_{𝔞}^i(M)$ is $𝔞$-cominimax for all $i<n$ if (and only if) ${\mathrm{\text{Hom}}}_R(R/𝔞,{\mathrm{\text{H}}}_{𝔞}^i(M))$ is minimax for all $i\le n$.

The notion of $f$-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper, we introduce the notion of $f$-biderivations as a natural “biderivation” counterpart of the notion of “$f$-derivations”. We first show, under some conditions, that any $f$-biderivation is a Jordan biderivation. Then, we turn to study $f$-biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibiderivation and an extremal biderivation. As a consequence, we show that every Jordan biderivation on a triangular algebra is a biderivation.

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial $P\in \mathbb{N}[x]$, and any directed finite bipartite graph can be considered as a polynomial $P\in \mathbb{N}[x,y]$, and vise verse. We also show that the multiplication in the semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$ corresponds to an operation of the corresponding graphs. This operation is exactly the product of Petri nets in the sense of Winskel [G. Winskel and M. Nielsen, Models of concurrency, in Handbook of Logic in Computer Science, Vol. 4, eds. Abamsky, Gabbay and Maibaum (Oxford University Press, 1995), pp. 1–148]. As an application, we give an approach to dividing in the semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$, and a criteria for parallalization of Petri nets. Finally, we endow the set of all bipartite graphs with the Zariski topology.

The algebraic structure of the group generated by the encryption functions of a block cipher depends on the key schedule algorithm used for generating the round keys. For such a reason, in general, studying this group does not appear to be an easy task. Previous works, focusing on the algebraic properties of groups associated to a cipher, have studied the group generated by the round functions of the cipher considering independent round keys.

In this paper, we want to study the more realistic group generated by the encryption functions, where the key schedule satisfies certain requirements. In this context, we are able to identify sufficient conditions that permit to guarantee the primitivity of this group and the security of the cipher with respect to the partition-based trapdoor. This type of trapdoor has been recently introduced by Bannier et al. in 2016 and it is a generalization of that introduced by Paterson in 1999.

We show that the ${{\mathscr{H}}}_{n-1}$-conjugacy classes of ${{\mathscr{H}}}_n,$ where ${{\mathscr{H}}}_n$ is the hyperoctahedral group on $2n$ elements, are indexed by marked bipartitions of $n.$ This will lead us to prove that $({{\mathscr{H}}}_n\times {{\mathscr{H}}}_{n-1},\mathrm{\text{diag}}({{\mathscr{H}}}_{n-1}))$ is a symmetric Gelfand pair and that the induced representation $1_{\mathrm{\text{diag}}({{\mathscr{H}}}_{n-1})}^{{{\mathscr{H}}}_n\times {{\mathscr{H}}}_{n-1}}$ is multiplicity free.

Crawley-Boevey introduced in [Poisson structure on moduli spaces of representations, J. Algebra 325 (2011) 205–215.], the notion of $H_0$-Poisson structure for associative algebras, which is the weakest condition that induces a Poisson structure on the moduli spaces of their representations. In this paper, by using a result of Armenta and Keller in [Derived invariance of the Tamarkin-Tsygan calculus of an algebra, C. R. Math. Acad. Sci. Paris 357(3) (2019) 236–240.], we show that an $H_0$-Poisson structure is preserved under derived Morita equivalence.

We give a relativization of the notions of essential, uniform, closed, injective and extending modules with respect to a left exact preradical on the ring. We also give useful examples and show that several classical properties hold for these kinds of modules.

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $({𝒜},{ℬ})$ in an exact category ${𝒞}$, ${𝒜}$ coincides with the class of relatively flat objects of ${𝒞}$ for some relative projectively generated exact structure, while ${ℬ}$ coincides with the class of relatively divisible objects of ${𝒞}$ for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.

For a commutative Noetherian ring $R$ of prime characteristic, denote by ${}^{f}R$ the ring $R$ with the left structure given by the Frobenius map. We develop Thomas Marley’s work on the property of the Frobenius functor $\mathrm{\text{F}}(-)=-{\otimes }_R{}^{f}R$ and show some interplays between $\mathrm{\text{F}}$ and its dual $\widetilde{\mathrm{\text{F}}}(-)={\mathrm{\text{Hom}}}_R{(}^{f}R-)$, which is introduced by Jürgen Herzog.