Journal of Algebra and Its Applications

The aim of the paper is to characterize the equation $1+\mathrm{\Delta }(R)=U(R)$, where $J(R)\subseteq \mathrm{\Delta }(R)=\{x\in R:x+u\in U(R)\mathrm{\text{for all}}u\in U(R)\}$ that is the largest Jacobson radical subring of $R$ and $U(R)$ is the set of invertible elements of a ring $R$. We show that this equation is closely related to $UJ$-rings and rings whose elements can be written as the sum of an idempotent and an element from $\mathrm{\Delta }(R)$. After presenting several characterizations and properties of this equation, we consider the rings satisfying the equation $1+\mathrm{\Delta }(R)=U(R)$ within many well-studied classes of rings. Finally, we close the paper with group rings.

A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Koşan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra 45 (2017) 2775–2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module $M$ over a semiprimary ring is nilpotent-invariant iff all nilpotent endomorphisms of submodules of $M$ extend to nilpotent endomorphisms of $M$. It is also shown that a right module $M$ over a prime right Goldie ring with dim$(M/Z(M))>1$ is nilpotent-invariant iff it is injective. We also study nilpotent-coinvariant modules that are the dual notation of nilpotent-invariant modules. It is proved that if $M$ is a finitely generated nilpotent-coinvariant right module with $M/J(M)$ square-full, then $M$ is quasi-projective. Some characterizations and structures of nilpotent-coinvariant modules are considered.

In this paper, we consider finite $2$-groups having the property that every non-abelian subgroup contains its centralizer. Such groups are classified completely up to isomorphism.

Here, we continue to characterize a recently introduced notion, le-modules ${}_{R}M$ over a commutative ring $R$ with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga. 158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec$(M)$ of all prime submodule elements of $M$. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec$(M)$ is connected if and only if R/Ann(M) contains no idempotents other than $\overline{0}$ and $\overline{1}$. Open sets in the Zariski topology for the quotient ring R/Ann(M) induces a base of quasi-compact open sets for the Zariski topology on Spec$(M)$. Every irreducible closed subset of Spec$(M)$ has a generic point. Besides, we prove a number of different equivalent characterizations for Spec$(M)$ to be spectral.

In this paper, we are mainly interested in the two questions “which are the commutative rings on which every finitely presented modules is $1$-periodic (respectively, $2$-periodic)?”. It is proved that these kinds of rings are particular cases of semiregular rings. So, we call them $1$-semiregular and $2$-semiregular rings, respectively. We establish characterizations of these rings in terms of various classical notions and we provide several examples of such rings.

Type B semigroups are generalizations of inverse semigroups, and every inverse semigroup admits an $E$-unitary cover (M. Petrich, Inverse Semigroups (Wiley, New York, 1984)). Motivated by studying $E$-unitary cover for inverse semigroups, and as a continuation of Petrich’s works in inverse semigroups, in this paper, we first introduce the concept of ∗-prehomomorphism of a type B semigroup. After obtaining some basic properties, we get some structure theorems and give some conditions for a type B semigroup which is constructed by using the ∗-prehomomorphism to be proper. In particular, we introduce the notion of $E$-unitary good cover for an abundant semigroup, and prove that every type B semigroup with compatible natural partial order admits an $E$-unitary good cover.

We prove that if $G$ is either a hypercentral-by-finite group or a soluble Baer group and if $G$ has finitely many non-isomorphic factor-groups, then $G$ is a Chernikov group. The converse is also true. Furthermore, we give some information on the structure of a metabelian group with finitely many non-isomorphic factor-groups.

The aim of this paper is twofold. First, to provide a characterization of clopen topologies. We show that a topology is clopen if and only if it is symmetric and Alexandroff. Second, to investigate when some special unitary subrings of a power set define a (clopen) topology.

A ring $R$ is said to be quasi-central Armendariz if $f(x)={\sum }_{i=0}^m{a}_i{x}^i$ and $g(x)={\sum }_{j=0}^n{b}_j{x}^j\in R[x]$ satisfy $f(x)g(x)=0$ then $R{a}_i{b}_j=a_i{b}_{j}R$ for all $i$ and $j$. It is proved that if $R$ is a quasi-central Armendariz ring then $f(x)g(x)=0$ implies that all $a_i{b}_j$ are in its Wedderburn radical $W(R)$, generalizing and improving the existing result in the literature.

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra $D^{\omega }(H)$ defined in [D. Bulacu and F. Panaite, A generalization of the quasi-Hopf algebra $D^{\omega }(G)$, Commun. Algebra 26 (1998) 4125–4141] and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for $D^{\omega }(H)$.

The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for $ℂ{ℳ}$, where ${ℳ}$ is any finite ${ℛ}$-trivial monoid. Their method relies on a technical result stating that ${ℛ}$-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an ${ℛ}$-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where ${ℒ}$-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for $\mathbb{Z}{ℳ}$, after which we prove that it also works for $R{ℳ}$ where $R$ is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if $R$ is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for $R{ℳ}$ is obtained from the one our algorithm yields for $\mathbb{Z}{ℳ}$ in a straightforward manner. In other words, for any finite ${ℛ}$-trivial monoid ${ℳ}$ our algorithm only has to be performed for $\mathbb{Z}{ℳ}$, after which a system of idempotents follows for any ring with a given system of idempotents.

In this paper, by taking the class of all $C3$ (or $D3$) right $R$-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via $D3$-covers (or $D3$-envelopes) and a right $V$-ring (or a right noetherian $V$-ring) via $C3$-covers (or $C3$-envelopes). By using isosimple-projective preenvelope, we obtained that if $R$ is a semiperfect right FGF ring (or left coherent ring), then every isosimple right $R$-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.

We study the deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base, which leads to describe the obstruction in extending a given deformation. We also discuss the construction of versal deformation of Courant pairs. As an application, we compute universal infinitesimal deformation of Poisson algebra structures on the three-dimensional complex Heisenberg Lie algebra. We compare the second deformation cohomology spaces of these Poisson algebra structures by considering them in the category of Leibniz pairs and Courant pairs, respectively.

A bounded poset $P:=(P,0,1,\le )$ is said to be lower finite if $P$ is infinite and for all $1\ne x\in P$, there are but finitely many $y\in P$ such that $y\le x.$ In this paper, we classify the modules $M$ over a commutative ring $R$ with identity with the property that the lattice $L_R(M)$ of $R$-submodules $M$ (under set-theoretic containment) is lower finite. Our results are summarized in Theorem 3.1 at the end of this note.

An element of a ring $R$ is said to be $r$-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring $R$ are $r$-precious, then $R$ is called an $r$-precious ring. We study some basic properties of $r$-precious rings. We also characterize von Neumann regular elements in $M_2(R)$ when $R$ is a Euclidean domain and by this argument, we produce elements that are $r$-precious but either not $r$-clean or not precious.

We introduce and study (strongly) $\pi$-Rickart objects and their duals in abelian categories, which generalize (strongly) self-Rickart objects and their duals. We establish general properties of such objects, we analyze their behavior with respect to coproducts, and we study classes all of whose objects are (strongly) $\pi$-Rickart. We derive consequences for module and comodule categories.

The aim of this paper is to define and study the involutive and weakly involutive quantum B-algebras. We prove that any weakly involutive quantum B-algebra is a residuated poset. As an application, we introduce and investigate the notions of existential and universal quantifiers on involutive quantum B-algebras. It is proved that there is a one-to-one correspondence between the quantifiers on weakly involutive quantum B-algebras. One of the main results consists of proving that any pair of quantifiers is a monadic operator on weakly involutive quantum B-algebras. We investigate the relationship between quantifiers on bounded sup-commutative pseudo BCK-algebras and quantifiers on other related algebraic structures, such as pseudo MV-algebras and bounded Wajsberg hoops.

Let $R$ be a unitary ring of finite cardinality $p^{\beta }k$, where $p$ is a prime number and $p\nmid k$. We show that if the group of units of $R$ has at most one subgroup of order $p$, then $R\cong A\oplus B,$ where $B$ is a finite ring of order $k$ and $A$ is a ring of cardinality $p^{\beta }$ which is one of the six explicitly described types.

We study the maximum Hamming distance (or rather, the complementary notion of “minimum approximability”) of a general function on a finite group $G$ to either of the sets End(G) and Aff(G), of group endomorphisms of $G$ and affine maps on $G$, respectively, the latter being a certain generalization of endomorphisms. We give general bounds on these two quantities and discuss an infinite class of extremal examples (where each of the two Hamming distances can be made as large as generally possible). Finally, we compute the precise values of the two quantities for all finite groups $G$ with $\left|G\right|\le 15$.

Large order cryptographically suitable quasigroups have important applications in the development of crypto-primitives and cryptographic schemes. These present new perspectives of cryptography and information security. From algebraic point of view polynomial completeness is one of the most important characteristic for cryptographically suitable quasigroups. In this paper, we propose four different methods to construct polynomially complete quasigroups of any order $n\ge 5$. First method is based on a starting quasigroup of same order, second method is based on a particular permutation of $S{S}_n$ and third and fourth methods are based on products of lower order quasigroups. In the last case, all quasigroups and their isotopes are polynomially complete. We also develop and implement an algorithm to derive a permutation for a given permutation of $S{S}_n,n\ge 5,$ so that they generate whole $S{S}_n$.

This paper is devoted to study 2-local derivations on $W$-algebra $W(2,2)$ which is an infinite-dimensional Lie algebra with some outer derivations. We prove that all 2-local derivations on the $W$-algebra $W(2,2)$ are derivations. We also give a complete classification of the 2-local derivation on the so-called thin Lie algebra and prove that it admits many 2-local derivations which are not derivations.

We use matrix wreath products to show that (1) every countable dimensional nonsingular algebra is embeddable in a finitely generated nonsingular algebra, (2) for every infinite dimensional finitely generated PI-algebra $A$ there exists an epimorphism $Â\stackrel{\phi }{\to }A$, where ${(ker\phi )}^3=(0)$ and the algebra $Â$ is not representable by matrices over a commutative algebra. If the algebra $A$ is commutative, then $Â$ satisfies the ACC on two-sided ideals as in the recent examples of Greenfeld and Rowen.

In this paper, we first give more interested properties of a weak Hopf quasigroup $H$. Next, we introduce the notion of Yetter–Drinfel’d weak quasimodule over $H$ and prove that the category ${{𝒲}{𝒬}{𝒴}{𝒟}}_H^H$ of right-right Yetter–Drinfel’d weak quasimodules is braided with left and right dualities under suitable conditions. Finally, we describe the notion of coquasitriangular weak Hopf quasigroup $(H,\sigma )$, and study there exist a relation between Yetter–Drinfel’d weak quasimodule and the coinvariant space of right $H$-comodule over $(H,\sigma )$.

Let $F$ be a field, and $F^n$ the $n$-dimensional row vector space over $F$. The orthogonality graph $O(F^n)$ of $F^n$ is an undirected simple graph which has ${(F^n)}^{\ast }=F^n\setminus \{0\}$ as its vertex set, and for distinct $\alpha ,\beta \in {(F^n)}^{\ast }$, $\alpha \sim \beta$ if and only if $\alpha {\beta }^T=0$, where ${\beta }^T$ is the transpose of $\beta$. When $F$ is finite, it is shown that any automorphism of $O(F^n)$ can be decomposed into the product of a row-orthogonal automorphism and either a permutation automorphism or a field automorphism; moreover, the fixing number and metric dimension of $O(F^n)$ are considered.