Journal of Algebra and Its Applications

L-algebras are based on an equation which is fundamental in the construction of various torsion-free groups, including spherical Artin groups, Riesz groups, certain mapping class groups, para-unitary groups, and structure groups of set-theoretic solutions to the Yang–Baxter equation. A topological study of L-algebras is initiated. A prime spectrum is associated to certain (possibly all) L-algebras, including three classes of L-algebras where the ideals are determined in a more explicite fashion. Known results on orthomodular lattices, Heyting algebras, or quantales are extended and revisited from an L-algebraic perspective.

We consider gradings on cluster algebras generated by $3\times 3$ skew-symmetric matrices. We show that, except in one particular case, mutation-cyclic matrices give rise to gradings in which all occurring degrees are positive and have only finitely many associated cluster variables. For mutation-acyclic matrices, we prove that all occurring degrees are associated with infinitely many variables. We also give a direct proof that the gradings are balanced in this case (i.e. that there is a bijection between the cluster variables of degree d and $-d$ for each occurring degree d).

Let ${𝒟}$ be either the category of R-modules or the category of chain complexes of R-modules and ${ℳ}$ a cofibrantly generated hereditary abelian model structure on ${𝒟}$. First, we get a new cofibrantly generated model structure on ${𝒟}$ related to ${ℳ}$ for any positive integer n, and hence, one can get new algebraic triangulated categories. Second, it is shown that any n-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let M be an R-module with finite flat dimension and n a positive integer, if $0\to M\to I_1\to I_2\to \cdots \to I_n\to M\to 0$ is an exact sequence of $R$-modules with every $I_i$ Gorenstein injective, then M is injective.

We introduce the definitions of g-derivations and generalized g-derivations on a ring R. The main objective of the paper is to describe the structure of a prime ring R in which g-derivations and generalized g-derivations satisfy certain algebraic identities with involution ∗, anti-automorphism and automorphism. Some well-known results concerning derivations, generalized derivations, skew derivations and generalized skew derivations in prime rings, have been generalized to the case of g-derivations and generalized g-derivations.

We show a necessary condition for Klein four symmetric pairs $(G,G^{\mathrm{\Gamma }})$ satisfying the condition (D.D.); that is, there exists at least one infinite-dimensional simple $(𝔤,K)$-module that is discretely decomposable as a $({𝔤}^{\mathrm{\Gamma }},K^{\mathrm{\Gamma }})$-module. This work is a continuation of [A criterion for discrete branching laws for Klein four symmetric pairs and its application to $E_{6(-14)}$, Int. J. Math. 31(6) (2020) 2050049]. Moreover, we define associated Klein four symmetric pairs, and we may use these tools to compute that a class of Klein four symmetric pairs do not satisfy the condition (D.D.); for example, $({𝔢}_{6(-14)},𝔰𝔬(4,2)\oplus 𝔰𝔬(2))$.

Let $S(D)$ represent a set of proper nonzero ideals $I(D)$ (respectively, t-ideals $I_t(D)$) of an integral domain $D\ne qf(D)$ and let P be a valid property of ideals of D. We say S(D) meets P (denoted $S(D)◃P)$ if each $s\in S(D)$ is contained in an ideal satisfying P. If S(D) $◃P,$ $dim(D)$ cannot be controlled. When $R=D[X],$ $I(D)$ $◃P$ does not imply I(R) $◃P$ while $I_t(D)$ $◃P$ implies $I_t(R)$ $◃P$ usually. We say S(D) meets P with a twist $($written $S(D){◃}^{t}P)$ if each $s\in S(D)$ is such that, for some $n\in N,$ $s^n$ is contained in an ideal satisfying P and study $S(D){◃}^{t}P,$ as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains.

In this paper, we introduce the notion of classical 2-prime subsemimodules of a semimodule M over a commutative semiring S with identity, which is a generalization of classical prime subsemimodules. A proper subsemimodule N of M having the property that for each $a,b\in S$ and each subsemimodule K of M, the inclusion $abK\subseteq N$ implies $a^{2}K\subseteq N$ or $b^{2}K\subseteq N$ is called classical 2-prime. We give some results concerning classical 2-prime subsemimodules. Also, the classical 2-prime avoidance theorem for subsemimodules is proved. Finally, the notion of valuation semimodules is introduced and some results on classical 2-prime subsemimodules in valuation semimodules are given.

For any triangular matrix coalgebra $\mathrm{\Gamma }=\left(\right.\begin{array}{cc}\hfill C\hfill & \hfill U\hfill \\ \hfill 0\hfill & \hfill D\hfill \end{array}\left)\right.$, in this paper, we first examine some connections between coalgebra properties of $\mathrm{\Gamma }$ and its constituent coalgebras C, D, which contain semiperfectness, computability and row/column-finiteness of their left Cartan matices. Then we devote to considering the coresolution dimensions of recollement of comodule categories by investigating covariantly finite subcategories.

Let p be a prime and let F be a finite field of characteristic p. Let FG denote the group algebra of the finite p-group G over the field F and let $V(FG)$ denote the group of normalized units in FG. The anti-automorphism $g\mapsto g^{-1}$ of G extends linearly to an anti-automorphism $a\mapsto a^{\ast }$ of FG. An element $u\in V(FG)$ is called unitary if $u^{\ast }=u^{-1}$. All unitary elements of $V(FG)$ form a subgroup which is denoted by $V_{\ast }(FG)$. If p is odd, the order of $V_{\ast }(FG)$ is $|F{|}^{(|G|-1)/2}$. However, to compute the order of $V_{\ast }(FG)$ still is open when $p=2$. In this paper, the order of $V_{\ast }(FG)$ is computed when G is a nonabelian $2$-group given by a central extension of the following form:

Let $\mathrm{\Gamma }$ be a torsionless grading monoid, $R={\oplus }_{\alpha \in \mathrm{\Gamma }}{R}_{\alpha }$ a $\mathrm{\Gamma }$-graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of $R_0$ and $G_0$ the group of units of $\mathrm{\Gamma }$. We say that R is a graded pseudo-valuation domain (gr-PVD) if whenever a homogeneous prime ideal P of R contains the product xy of two homogeneous elements of $R_H$, then $x\in P$ or $y\in P$. The notion of gr-PVDs was introduced recently by the authors in (M. T. Ahmed, C. Bakkari, N. Mahdou and A. Riffi, Graded pseudo-valuation domains, Comm. Algebra 48 (2020) 4555–4568) as a graded version of pseudo-valuation domains (PVDs). In this paper, we show that R is a gr-PVD if and only if exactly one of the following two conditions holds :

• ⁽¹⁾
  • ^(a)$R_0=K$,
  • ^(b)$\mathrm{\Gamma }$ is a pseudo-valuation monoid, and
  • ^(c)$R_{\alpha }={(R_H)}_{0}x$ for every $0\ne x\in R_{\alpha }$ whenever $\alpha \in \mathrm{\Gamma }$ is not a unit.
• ⁽²⁾
  • ^(a)$R_0\ne K$,
  • ^(b)$\mathrm{\Gamma }$ is a valuation monoid,
  • ^(c)$R_{\alpha }=Kx$ for every $0\ne x\in R_{\alpha }$ whenever $\alpha \in \mathrm{\Gamma }$ is not a unit, and
  • ^(d)$T={\oplus }_{\alpha \in G_0}{R}_{\alpha }$ is a gr-PVD.

Let D be an integral domain and w be the so-called w-operation on D. In this paper, we define the notion of w-ZPUI domains which is a natural generalization of ZPUI domains introduced by Olberding in 2000. We say that D is a w-ZPUI domain if every nonzero proper w-ideal I of D can be written as $I={(J{P}_1\cdots P_n)}_w$ for some w-invertible ideal J of D and $\{P_1,\dots ,P_n\}$ is a nonempty collection of pairwise w-comaximal prime w-ideals of D. Then, among other things, we show that D is a w-ZPUI domain if and only if the polynomial ring $D[X]$ is a w-ZPUI domain, if and only if D is a strongly discrete independent ring of Krull type. We construct three types of new w-ZPUI domains from an old one by $A+XB[X]$-construction, pullback, and $D[\{X_i,Y_i,U_i,V_i\}]/(\{X_i{V}_i-Y_i{U}_i\})$-domains. We also show that given an abelian group G, there is a ZPUI domain with ideal class group G but not a Dedekind domain. Finally, we introduce and study the notion of w-ISP domains as a generalization of w-ZPUI domains.

Let V be a valuation domain of rank one with quotient field K. We study the set of extensions of V to the field of rational functions K(X) induced by pseudo-convergent sequences of K from a topological point of view, endowing this set either with the Zariski or with the constructible topology. In particular, we consider the two subspaces induced by sequences with a prescribed breadth or with a prescribed pseudo-limit. We give some necessary conditions for the Zariski space to be metrizable (under the constructible topology) in terms of the value group and the residue field of V.

Groups associated to surfaces isogenous to a higher product of curves can be characterized by a purely group-theoretic condition, which is the existence of the so-called ramification structure. In this paper, we prove that infinitely many quotients of the Grigorchuk groups admit ramification structures. This gives the first explicit infinite family of 3-generated finite 2-groups with ramification structures.

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, G be a finite group and $\sigma (G)=\{{\sigma }_i|{\sigma }_i\cap \pi (G)\ne \varnothing \}$. A set ${\mathscr{H}}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of G if every non-identity member of ${\mathscr{H}}$ is a Hall ${\sigma }_i$-subgroup of G for some i and ${\mathscr{H}}$ contains exactly one Hall ${\sigma }_i$-subgroup of G for every ${\sigma }_i\in \sigma (G)$. Let ${\mathscr{H}}$ be a complete Hall $\sigma$-set of G. A subgroup A of G is said to be $\sigma$-semipermutable with respect to ${\mathscr{H}}$ if $A{H}^x=H^{x}A$ for all $x\in G$ and all $H\in {\mathscr{H}}$ such that $(|H|,|A|)=1$; $\sigma$-semipermutable in G if A is $\sigma$-semipermutable in G with respect to some complete Hall $\sigma$-set of G. We say that a subgroup H of G is weakly $\sigma$-semipermutable in G if there exists a $\sigma$-permutable subgroup $T$ of $G$ such that HT is $\sigma$-permutable in G and $H\cap T\le H_{\sigma sG}$, where $H_{\sigma sG}$ is the subgroup of H generated by all those subgroups of H which are $\sigma$-semipermutable in G. In this paper, we study the structure of G under the condition that some subgroups of G are weakly $\sigma$-semipermutable in G.

Let K be a field of characteristic zero and $X_n=\{x_1,\dots ,x_n\}$ be a finite set of variables. Consider the free metabelian Poisson algebra $P_n$ of rank n generated by $X_n$ over K. An element in $P_n$ is called symmetric if it is preserved under any change of variables, i.e. under the action of each permutation in $S_n$. In this study, we determine the algebra $P_n^{S_n}$ of symmetric polynomials of $P_n$.

Various authors have been generalizing some unital ring properties to nonunital rings. We consider properties related to cancellation of modules (being unit-regular, having stable range one, being directly finite, exchange, or clean) and their “local” versions. We explore their relationships and extend the defined concepts to graded rings. With graded clean and graded exchange rings suitably defined, we study how these properties behave under the formation of graded matrix rings. We exhibit properties of a graph E which are equivalent to the unital Leavitt path algebra $L_K(E)$ being graded clean. We also exhibit some graph properties which are necessary and some which are sufficient for $L_K(E)$ to be graded exchange.

In this paper, we study a new flexible method to extend linearly the graph of a nonlinear, and usually not bijective, function so that the resulting extension is a bijection. Our motivation comes from cryptography. Examples from symmetric cryptography are given as how the extension was used implicitly in the construction of some well-known block ciphers. The method heavily relies on ideas brought from linear coding theory and secret sharing. We are interested in the behavior of the composition of many extensions, and especially the space of parameters that defines a family of equations based on finite differences or linear forms. For any linear extension, we characterize entirely the space of parameters for which such equations are solvable in terms of the space of parameters that render those equations for the corresponding nonlinear extended functions solvable. Conditions are derived to assess the solvability of those kind of equations in terms of the number of compositions or iterations. We prove a relation between the number of compositions and the dimensions of vector spaces that appear in our results. The proofs of those properties rely mostly on tools from linear algebra.

Let $\text{Vect}(ℝ)$ be the Lie algebra of smooth vector fields on $ℝ$ and ${{𝒟}}_{\lambda ,\mu ;\nu }:={\text{Hom}}_{\text{diff}}({{ℱ}}_{\lambda }\otimes {{ℱ}}_{\mu },{{ℱ}}_{\nu })$ be the space of bilinear differential operators acting on weighted densities. In this paper, we classify $𝔰𝔩(2)$-invariant skew-symmetric binary differential operators from $\text{Vect}(ℝ)\wedge \text{Vect}(ℝ)$ to ${{𝒟}}_{\lambda ,\mu ;\nu }$ vanishing on $𝔰𝔩(2)$. This result allows us to compute the second differential $𝔰𝔩(2)$-relative cohomology of $\text{Vect}(ℝ)$ with coefficients in ${{𝒟}}_{\lambda ,\mu ;\nu }$.

In this paper, we focus on the initial degree and the vanishing of the Valabrega–Valla module of a pair of monomial ideals $J\subseteq I$ in a polynomial ring over a field $𝕂$. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of J. For edge ideal of graphs, a complete characterization of the vanishing of the Valabrega–Valla module is given. For higher degree ideals, we find classes, where the Valabrega–Valla module vanishes. For the case that J is the facet ideal of a clutter ${𝒞}$ and I is the defining ideal of singular subscheme of J, the non-vanishing of this module is investigated in terms of the combinatorics of ${𝒞}$. Finally, we describe the defining ideal of the Rees algebra of I/J provided that the Valabrega–Valla module is zero.

In this paper, we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture, we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyominoes viewed as generalizations of closed paths.

We describe skew G-codes, which are codes that are the ideals in a skew group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. These codes generalize many of the well-known classes of codes such as cyclic, quasicyclic, constacyclic codes, skew cyclic, skew quasicyclic and skew constacyclic codes. Additionally, using the skew G-matrices, we can generalize almost all the known constructions in the literature for self-dual codes.