Journal of Algebra and Its Applications

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. $R$ is said to be weakly clean if each element of $R$ is either a sum or a difference of a unit and an idempotent, and $R$ is said to be feebly clean if every element $r$ can be written as $r=u+e_1-e_2$, where $u$ is a unit and $e_1,e_2$ are orthogonal idempotents. Clearly, clean rings are weakly clean rings and both of them are feebly clean. In a recent paper (J. Algebra Appl. 17 (2018) 1850111 (5 pp.)), McGoven characterized when the group ring ${\mathbb{Z}}_{(p)}[C_q]$ is weakly clean and feebly clean, where $p,q$ are distinct primes. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, ${𝒪}$ its ring of integers, $𝔭\subset {𝒪}$ a nonzero prime ideal and ${{𝒪}}_{𝔭}$ the localization of ${𝒪}$ at $𝔭$. We investigate when the group ring ${{𝒪}}_{𝔭}[G]$ is weakly clean and feebly clean, where $G$ is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when $K=\mathbb{Q}({\zeta }_n)$ is a cyclotomic field or $K=\mathbb{Q}(\sqrt{d})$ is a quadratic field.

We classify all graded compatible left-symmetric algebraic structures on high rank Witt algebras, and classify all non-graded ones satisfying a minor condition. Furthermore, graded compatible left-symmetric algebraic structures on high rank Virasoro algebras are also classified.

Let $D$ be an integral domain with quotient field $K$, $R$ be an overring of $D$, $X$ be an indeterminate over $R$ and $E$ be a subset of $K$. We consider the ring of $D$-valued$R$-polynomials on$E$, denoted by ${\mathrm{\text{Int}}}_R(E,D)$, formed by the polynomials $f\in R[X]$ such that $f(a)\in D$ for each $a\in E$, that is, ${\mathrm{\text{Int}}}_R(E,D):=\{f\in R[X]:f(E)\subseteq D\}$. In this paper, we study localization properties, local freeness and faithful flatness of ${\mathrm{\text{Int}}}_R(E,D)$ over various classes of locally essential domains. In particular, we extend some known results to more general settings.

The purpose of this paper is to give some sufficient and necessary conditions for an element to be a strongly $EP$ element through some transformations of equations.

We introduce and study a class of objects that encompasses Christensen and Foxby’s semidualizing modules and complexes and Kubik’s quasi-dualizing modules: the class of $𝔞$-adic semidualizing modules and complexes. We give examples and equivalent characterizations of these objects, including a characterization in terms of the more familiar semidualizing property. As an application, we give a proof of the existence of dualizing complexes over complete local rings that does not use the Cohen Structure Theorem.

The notion of maximal non-$\varphi$-pseudo-valuation ring is introduced which generalizes the concept of maximal non-pseudo-valuation domain. The equivalence of maximal non-$\varphi$-PVR and maximal non-local ring is established under some conditions. The concept of maximal non-PVR is also introduced and it is shown that a maximal non-PVR is either a $\varphi$-PVR or a maximal non-$\varphi$-PVR. Examples are also given to show that all the introduced concepts are independent of each other, in general.

The second cohomology group (SCG) of the Jordan superalgebra ${{𝒟}}_t$, $t\ne 0$, over an algebraically closed field $𝔽$ of characteristic zero is calculated by using the coefficients which appear in the regular superbimodule $\mathrm{\text{Reg}}{{𝒟}}_t$. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem [F. A. Gómez-González, Wedderburn principal theorem for Jordan superalgebras I, J. Algebra 505 (2018) 1–32]. First, to calculate the SCG of a Jordan superalgebra we use split-null extension of the Jordan superalgebra and the Jordan superalgebra representation. We prove conditions that satisfy the bilinear forms $h$ that determine the SCG in Jordan superalgebras. We use these to calculate the SCG for the Jordan superalgebra ${{𝒟}}_t$, $t\ne 0$. Finally, we prove that ${{\mathscr{H}}}^2({{𝒟}}_t,\mathrm{\text{Reg}}{{𝒟}}_t)=0\dot{+}{𝔽}^2$, $t\ne 0$.

Let $R$ be a commutative ring. A polynomial $f\in R[X]$ is an annihilating content (AC) polynomial if $f=ag$ where $a\in R$ and $g\in R[X]$ is a nonzerodivisor and $R$ is an EM-ring if each $f\in R[X]$ is an AC polynomial. In this paper, we investigate AC polynomials and EM-rings and extensions to modules. For example, we show that $R$ is an EM-ring if and only if every linear polynomial is an AC polynomial if and only if for each finitely generated ideal $I$ of $R$, $I=aJ$ where $a\in R$ and $J$ is a finitely generated ideal of $R$ with $(0:J)=0$. Special attention is given to the case when $R$ is Noetherian where such rings are characterized by the property that each associated prime is principal.

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is an ${\mathscr{H}}$-subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$; $H$ is called weakly ${\mathscr{H}}C$-embedded in $G$ if $G$ has a normal subgroup $T$ such that $H^G=HT$ and $N_T(H)\cap H^g\le H$ for all $g\in G,$ where $H^G$ is the normal closure of $H$ in $G$. In this paper, we study the $p$-nilpotence of a group $G$ in which every subgroup of order $d$ of a Sylow $p$-subgroup $P$ with $1<d<\left|P\right|$ is weakly ${\mathscr{H}}C$-embedded in $G$. Many recent results in the literature related to $p$-nilpotence of $G$ are generalized.

In this paper, we determine a family ${𝔓}_{\Re }^H$ of subsets of a hypergroup $H$ such that the geometric space $(H,{𝔓}_{\Re }^H)$ is strongly transitive. We show that if $(H,\circ )$ is a hypergroup and $\Re$ is strongly regular relation then ${\rho }_{\Re }$ is an equivalence relation.

Let $R$ be a ring, $S$ a strictly ordered monoid and $\omega :S\to \mathrm{\text{End}}(R)$ a monoid homomorphism. In this paper, we investigate the problem when a skew generalized power series ring $R[[S,\omega ]]$ has the same triangulating dimension as the ring $R$. Furthermore, for a piecewise prime ring we determine a large class of the skew generalized power series ring which have a generalized triangular matrix representation for which the diagonal rings are prime. Our results extend and unify many existing results.

Let $\pi$ be a group and $H=({\{H_{\alpha }\}}_{\alpha \in \pi },\mathrm{\Delta },𝜀,S,\mathrm{\Phi })$ a quasi-Turaev group coalgebra over $\pi$. In this paper, we firstly construct the category ${𝒴}{𝒟}(H)$ of left-right Yetter–Drinfeld modules over $H$, generalizing both of the Yetter–Drinfeld modules over Hopf group coalgebra and quasi-Hopf algebra, and prove that this category is isomorphic to the center of the representation category of $H$. Next, we prove that the full subcategory ${𝒴}{𝒟}{(H)}^{\mathrm{\text{fd}}}$ consisting of all finite dimensional Yetter–Drinfeld modules over $H$ is autonomous. Finally, when $H$ is reduced to a Turaev group coalgebra, we introduce the dual 2-cocycle $\mathrm{\Theta }$ of $H$, and dual cocycle deformation $H^{\mathrm{\Theta }}$, and show that ${𝒴}{𝒟}(H)$ is isomorphic to ${𝒴}{𝒟}(H^{\mathrm{\Theta }})$ as braided monoidal category.

Based on the concept of a lattice preradical recently introduced in [T. Albu and M. Iosif, Lattice preradicals with applications to Grothendieck categories and torsion theories, J. Algebra 444 (2015) 339–366], we present and investigate in this paper, the latticial counterparts of the concepts of prime and irreducible preradicals on the category Mod-$R$ of all unital right $R$-modules over an associative ring $R$ with identity, introduced and studied in [F. Raggi, J. Ríos Montes, H. Rincón, R. Fernández-Alonso and C. Signoret, Prime and irreducible preradicals, J. Algebra Appl. 4 (2005) 451–466].

In [S. Benayadi, F. Mhamdi and S. Omri, Quadratic (resp. symmetric) Leibniz superalgebras, Commun. Algebra, doi:10.1080/00927872.2020.1850751], the quadratic Leibniz superalgebras, which are (left or right) Leibniz superalgebras provided with even supersymmetric non-degenerate and associative bilinear forms, were investigated. Besides the notion of associativity, there are other kinds of invariance for bilinear forms on Leibniz superalgebras, namely the left invariance and the right invariance. In this paper, we investigate the Leibniz superalgebras endowed with even, supersymmetric, non-degenerate and left invariant bilinear forms and highlight the links between these superalgebras and some other algebraic structures. More precisely, every symmetric Leibniz superalgebra provided with such a bilinear form gives rise to a new type of superalgebra which we call LS-Lie superalgebra. We study LS-Lie superalgebras and we give some interesting informations on the structure of these superalgebras by using certain extensions introduced in [S. Benayadi and F. Mhamdi, Odd-quadratic Leibniz superalgebras, Adv. Pure Appl. Math. 10(4) (2019) 287–298; S. Benayadi, F. Mhamdi and S. Omri, Quadratic (resp. symmetric) Leibniz superalgebras, Commun. Algebra, doi:10.1080/00927872.2020.1850751]. Further, several nontrivial examples of LS-Lie superalgebras are included. Finally, we give similar results for Leibniz superalgebras with right invariant bilinear forms.

Let ${ℰ}$ be an injectively resolving subcategory of left $R$-modules. We introduce and study ${ℰ}$-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the ${ℰ}$-Gorensteinflat modules, investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, stability of the category of ${ℰ}$-Gorensteinflat modules is discussed, and some known results are obtained as applications.

In this paper, we focus on discussing diagonal solutions and general solutions of second-order matrix polynomial equation of high degree in complex field. By characterizing some algebraic properties of the mentioned two types of the solutions, we present sufficient conditions that a general second-order matrix polynomial equation has diagonal solutions or general solutions. Analytic expressions of the solutions, as well as the corresponding algorithms for finding the solutions are provided. An example is given so as to verify the theoretical results we have derived.

In this paper, we deal with $(m,n)$-hyperideals of ordered semihyperrings. First, we provide some properties about these $(m,n)$-hyperideals. Next, we provide nontrivial examples of $(m,n)$-hyperideals that are not left/right hyperideals. Finally, we apply our results to ordered semirings and provide nontrivial examples on $(m,n)$-ideals that are not left/right ideals.

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson–Badawi ${\omega }_{R[X]}(I[X])={\omega }_R(I)$ conjecture, we prove that under some conditions, the formula ${\omega }_B(I^{𝜖})={\omega }_R(I)$ holds for some amount $R$-algebras $B$ and some ideals $I$ of $R$, where ${\omega }_R(I)$ is the smallest positive integer $n$ that the ideal $I$ of $R$ is $n$-absorbing. A corollary to the mentioned formula is that if, for example, $R$ is a Prüfer domain or a torsion-free valuation ring and $I$ is a radical ideal of $R$, then ${\omega }_{R[[X]]}(I[[X]])={\omega }_R(I)$.

In this work, we present the integral trace form ${\mathrm{\text{Tr}}}_{𝕄/\mathbb{Q}}(x^2)$ of a cyclic extension $𝕄/\mathbb{Q}$ with degree $pq$, where $𝕄=𝕂𝕃$, $p$ and $q$ are distinct odd primes, the conductor of $𝕄$ is a square free integer, and $x$ belongs to the ring of algebraic integers ${{𝒪}}_{𝕄}$ of $𝕄$. The integral trace form of $𝕄/\mathbb{Q}$ allows one to calculate the packing radius of lattices constructed via the canonical (or twisted) homomorphism of submodules of ${{𝒪}}_{𝕄}$.

We continue a very fruitful line of inquiry into the multiplicative ideal theory of an arbitrary Leavitt path algebra $L$. Specifically, we show that factorizations of an ideal in $L$ into irredundant products or intersections of finitely many prime-power ideals are unique, provided that the ideals involved are powers of distinct prime ideals. We also characterize the completely irreducible ideals in $L$, which turn out to be prime-power ideals of a special type, as well as ideals that can be factored into products or intersections of finitely many completely irreducible ideals.

Let ${{𝒪}{𝒫}{ℰ}}_n$ be the monoid of all orientation-preserving and extensive full transformations on $\{1,2,\dots ,n\}$ ordered in the standard way. In this paper, we determine the minimum generating set and the minimum idempotent generating set of ${{𝒪}{𝒫}{ℰ}}_n$, and so the rank and the idempotent rank of ${{𝒪}{𝒫}{ℰ}}_n$ are obtained. Moreover, we describe maximal subsemigroups and maximal idempotent generated subsemigroups of ${{𝒪}{𝒫}{ℰ}}_n$ and completely obtain their classifications.