Journal of Algebra and Its Applications

For a finite group $G$, we associate the quantity $\beta (G)=\frac{|L(G)|}{\left|G\right|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of $\beta$ on the class of $p$-groups of order $p^n$, where $n\ge 3$ is an integer. We show that the set containing the quantities $\beta (G)$, where $G$ is a finite (abelian) group, is dense in $[0,\infty ).$ Finally, we consider $\beta$ to be a function on $L(G)$ and we indicate some of its properties, the main result being the classification of finite abelian $p$-groups $G$ satisfying $\beta (H)\le 1,\forall H\in L(G).$

In this paper, we construct planar polynomials of the type $f_{A,B}(x)=x(x^{q^2}+A{x}^q+Bx)\in {𝔽}_{q^3}[x]$, with $A,B\in {𝔽}_q$. In particular, we completely classify the pairs $(A,B)\in {𝔽}_q^2$ such that $f_{A,B}(x)$ is planar using connections with algebraic curves over finite fields.

Let $\mathrm{\Lambda }$ be an algebra with an indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support $\tau$-tilting $\mathrm{\Lambda }$-modules. In this paper, we will show that it can be restricted to $\tau$-tilting modules.

Not every element in a lattice has a complement. In this paper we introduce a notion of ranked complement, which depends on a natural number $n$, so that for every element $x$ in a lattice with finite height there exists $n$ such that $x$ has a complement of rank $n$. One of the main results we establish is that in a modular lattice having finite height, every element has a complement of rank less than $n+1$ if and only if there is a chain $0=a_0\le a_1\le \cdots \le a_n=1$ of elements such that each interval $[a_i,a_{i+1}]$ is a complemented lattice.

In this paper, we consider Hom-Lie groups and introduce left invariant almost contact structures on them (almost contact Hom-Lie algebras). On such Hom-Lie groups, we construct the almost contact metrics and the contact forms. We give the notion of normal almost contact Hom-Lie algebras and describe $K$-contact and Sasakian structures on Hom-Lie algebras. Also, we study some of their properties. In addition, it is shown that any Sasakian Hom-Lie algebra is a $K$-contact Hom-Lie algebra. Finally, we present examples of Sasakian Hom-Lie algebras and in particular, we show that the skew symmetric matrix $𝔰𝔬(3)$ carries a Sasakian structure.

Let $Ḡ$ be the simple algebraic supergroup $\mathrm{\text{SL}}(m|n)$ or $𝔬𝔰𝔭(m|2n)$ over $ℂ$. Let $𝔤=\mathrm{\text{Lie}}(Ḡ)={𝔤}_{\bar{0}}\oplus {𝔤}_{\bar{1}}$ and let $G=Ḡ(ℂ)$, where $ℂ$ is considered as a superalgebra concentrated in even degree. Suppose $e\in {𝔤}_{\bar{0}}$ is nilpotent. We describe the centralizer ${𝔤}^e$ of $e$ in $𝔤$ and its center $𝔷({𝔤}^e)$. In particular, we give bases for ${𝔤}^e$, $𝔷({𝔤}^e)$ and ${(𝔷({𝔤}^e))}^{G^e}$. We also determine the labeled Dynkin diagram $\mathrm{\Delta }$ with respect to $e$ and subsequently describe the relation between ${(𝔷({𝔤}^e))}^{G^e}$ and $\mathrm{\Delta }$.

A right $R$-module $M$ is called a $U$-module if, whenever $A$ and $B$ are submodules of $M$ with $A\cong B$ and $A\cap B=0,$ there exist two direct summands $K$ and $T$ of $M$ such that $A{\subseteq }^{\mathrm{\text{ess}}}K$, $B{\subseteq }^{\mathrm{\text{ess}}}T$ and $K\oplus T{\subseteq }^{\oplus }M$. The class of $U$-modules is a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper we study the rings whose cyclics are $U$-modules, extending many of the known results on the subject and providing new ones.

Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek–Geiringer–Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 4.2 and its corresponding Algorithm 1 which decide if a configuration is $𝜀$-locally rigid, a notion we define. A configuration which is $𝜀$-locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius $𝜀$ in configuration space. Deciding $𝜀$-local rigidity is possible for configurations which are smooth or singular, generic or non-generic. We also present Algorithms 2 and 3 which use numerical algebraic geometry to compute a discrete-time sample of a continuous flex, providing useful visual information for the scientist.

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from $M_n(ℂ)$, the set of all $n\times n$ matrices over $ℂ$, to the set ${{ℛ}}^{\text{ⓓ}}$ of all core-EP invertible elements in an arbitrary unital ring ${ℛ}$ with involution. A new partial order is introduced on ${{ℛ}}^{\text{ⓓ}}$ by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.

A notion of $n$-cotorsion pairs in an extriangulated category with enough projectives and enough injectives is defined in this paper. We show that there exists a one-to-one correspondence between $n$-cotorsion pairs and $(n+1)$-cluster tilting subcategories. As an application, this result generalizes the work by Huerta, Mendoza and Pérez in an abelian case. Finally, we give some examples illustrating our main result.

In this work, the notions of a partial action of a weak Hopf algebra on a coalgebra and of a partial action of a groupoid on a coalgebra will be introduced, together with some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of a partial action on a coalgebra and of a partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.

An idempotent in a ring is called fine (see G. Călugăreanu and T. Y. Lam, Fine rings: A new class of simple rings, J. Algebra Appl. 15(9) (2016) 18) if it is a sum of a nilpotent and a unit. A ring is called an idempotent-fine ring (briefly, an $IF$ ring) if all its nonzero idempotents are fine. In this paper, the properties of $IF$ rings are studied. A notable result is proved: The diagonal idempotents $E_{11}+\cdots +E_{ii}$ ($i=1,\dots ,n$) are fine in the matrix ring ${{ℳ}}_n(R)$ for any unital ring $R$ and any positive integer $n$. This yields many classes of rings over which matrix rings are $IF$.

Starting with the two semi-direct products of two copies of the group of integers, a description of all possible semi-direct products of these two groups with finite groups, under a faithful action, is obtained. The results are groups that are isomorphic to 16 crystallographic groups in dimension two. An extra effort gives us a group isomorphic to the 17th crystallographic group. Four of the crystallographic groups are shown to be the only solutions to a natural generalization of the semi-direct product of two copies of the group of integers. Finally, we verify that the two additional groups, that are solution to the generalization of the semi-direct product of two copies of the integers, preserve stability in the sense that their semi-direct products with finite groups under a faithful action give rise only to crystallographic groups, if we disregard some trivial outcomes.

We prove a determinantal type formula to compute the characters of a class of finite-dimensional irreducible representations of the general Lie super-algebra $𝔤𝔩(m|n)$ in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula, originally conjectured by van der Jeugt and Moens, can be regarded as a generalization of the well-known Jacobi–Trudi formula.

Let $S$ be a commutative ring, $x,y\in S$ a pair of exact zero divisors, and $R=S/(x)$. Let $F$ be a complex of free $R$-modules. In this paper we explicitly compute cohomological operators of $R$ over $S$ by constructing endomorphisms of $F$. We consider some properties of these cohomological operators, as well as provide an example in which these cohomological operators act non-trivially.

We consider Anderson $t$-motives $M$ of dimension 2 and rank 4 defined by some simple explicit equations parameterized by $2\times 2$ matrices. We use methods of explicit calculation of $h^1(M)$ — the dimension of their cohomology group $H^1(M)$ ($=\mathrm{\text{the}}$ dimension of the lattice of their dual $t$-motive $M^{\prime }$) developed in our earlier paper. We calculate $h^1(M)$ for $M$ defined by all matrices of the form $\left(\begin{array}{cc}\hfill 0\hfill & \hfill a_{12}\hfill \\ \hfill a_{21}\hfill & \hfill 0\hfill \end{array}\right)$, and by some matrices of the form $\left(\begin{array}{cc}\hfill a_{11}\hfill & \hfill a_{12}\hfill \\ \hfill a_{21}\hfill & \hfill 0\hfill \end{array}\right)$. These methods permit to make analogous calculations for most (probably all) $t$-motives. $h^1$ of all Anderson $t$-motives $M$ under consideration satisfy the inequality $h^1(M)\le 4$, while in all known examples we have $h^1(M)=0,1,4$. Do there exist $M$ of this type having $h^1=2,3$? We do not know, this is a subject of further research.

We study $s$-arc-transitive cubic graph $\mathrm{\Gamma }$, and give a characterization of minimal normal subgroups of the automorphism group. In particular, each $\mathrm{\Gamma }$ with quasi-primitive automorphism group is characterized. An interesting consequence of this characterization states that a non-solvable minimal normal subgroup $M$ contains at most 2 copies of non-abelian simple group when it acts transitively on arcs, or contains at most 6 copies of non-abelian simple group when it acts regularly on vertices.

Let $A$, $R$ be two finite dimensional algebras over a field $k$, such that $R$ is a split extension of A by the nilpotent bimodule $Q$. We mainly give necessary and sufficient conditions for a tilting pair $(C,T)$ such that $(C{\otimes }_{A}R,T{\otimes }_{A}R)$ or $(C{\otimes }_{R}A,T{\otimes }_{R}A)$ are tilting pairs. Also, we obtain a similar condition such that a Wakamatsu tilting pair $(C,T)$ in $A$-mod can be a Wakamatsu tilting pair $(C{\otimes }_{A}R,T{\otimes }_{A}R)$ in $R$-mod.

It is well known that Frobenius groups can be defined by their complement subgroups. But until now we cannot use a complement subgroup to define a modular Frobenius group. In the present paper, a generalization of Frobenius complements is used as a characterization of a class of modular Frobenius groups. In fact, we build a connection between modular Frobenius groups and Frobenius–Wielandt groups.

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no nonzero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so-called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the $e$-injective semimodule, and the $i$-injective semimodules through several implications, examples and counter examples. Moreover, we show that every semimodule over an arbitrary semiring can be embedded in a $c$-$i$-injective semimodule.

Let $M$ be a torsion-free module over an integral domain $D$ with quotient field $K$. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module $M$ is a $v$-multiplication module if and only if ${(𝔭M)}_v$ is a maximal $v$-submodule and ${({𝔭}^{-1}𝔭)}_{v_1}=D$ for every minimal prime ideal $𝔭$ of $D$. If $M$ is a finitely generated Krull module, then $M_1=\cap M_{𝔭}$ is a Krull module and $v$-multiplication module. It is also shown that the following three conditions are equivalent: $M$ is completely integrally closed, $M[x]$ is completely integrally closed, and $M[[x]]$ is completely integrally closed.