Journal of Algebra and Its Applications

In this paper, for any integer $n\ge 1$ we introduce a non-metacyclic finite group of order $3^{3n+2}$ and nilpotency class $2n+1$ with zero deficiency.

Using the equivalence of unique generation and cleanness of $C(X)$, we give affirmative answers to questions raised in [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc.  66 (1949) 464–491] and [D.D. Anderson et al., When are associates unit multiples? Rocky Mountain J. Math.  34 (2004) 811–828] for rings of real-valued continuous functions. In fact, we show that if $C(X)$ is $\mathrm{\text{UG}}$ (uniquely generated) then $M_n(C(X))$ is too, and $R$ is strongly associate if and only if $R[[x]]$ is, where $R=C(X)$. We give topological characterizations of $\mathrm{\text{UG}}$ and $\mathrm{\text{AS}}$ (annihilator-stable) elements of $C(X)$ for continuum spaces $X$ and using this, we observe that the product of two UG elements need not be $\mathrm{\text{UG}}$. It is shown that the set of elements in $C(X)$ which have stable range 1 and the set of $\mathrm{\text{AS}}$ elements of $C(X)$ coincide and several examples are given which show that the set of $\mathrm{\text{UG}}$ elements, the set of $\mathrm{\text{AS}}$ elements and the set of clean elements of $C(X)$ can differ. Finally, we characterize spaces $X$ for which every clean element of $C(X)$ or every element of $C(X)$ which has stable range 1 is $\mathrm{\text{UG}}$.

Let $R$ be a commutative ring with $1\ne 0$. Recall that a proper ideal $I$ of $R$ is called a 2-absorbing ideal of $R$ if $a,b,c\in R$ and $abc\in I$, then $ab\in I$ or $ac\in I$ or $bc\in I$. A more general concept than 2-absorbing ideals is the concept of $n$-absorbing ideals. Let $n\ge 1$ be a positive integer. A proper ideal $I$ of $R$ is called an n-absorbing ideal of $R$ if $a_1,a_2,\dots ,a_{n+1}\in R$ and $a_1,a_2\cdots a_{n+1}\in I$, then there are $n$ of the $a_i$’s whose product is in $I$. The concept of $n$-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of $R$ is a 1-absorbing ideal of $R$). Let $m$ and $n$ be integers with $1\le n<m$. A proper ideal $I$ of $R$ is called an $(m,n)$-closed ideal of $R$ if whenever $a^m\in I$ for some $a\in R$ implies $a^n\in I$. Let $A$ be a commutative ring with $1\ne 0$ and $M$ be an $A$-module. In this paper, we study $n$-absorbing ideals and $(m,n)$-closed ideals in the trivial ring extension of $A$ by $M$ (or idealization of $M$ over $A$) that is denoted by $A(+)M$.

The class of Pascal finite polynomial automorphisms is a subclass of the class of locally finite ones allowing a more effective approach. In characteristic zero, a Pascal finite automorphism is the exponential map of a locally nilpotent derivation. However, Pascal finite automorphisms are defined in any characteristic, and therefore constitute a generalization of exponential automorphisms to positive characteristic. In this paper, we prove several properties of Pascal finite automorphisms. We obtain in particular that the Pascal finite property is stable under taking powers but not under composition. This leads us to formulate a generalization of the exponential generators conjecture to arbitrary characteristic.

We recover the holomorphic discrete series representations of $SU(1,n)$ as well as some unitary irreducible representations of $SU(n+1)$ by deformation of a minimal realization of $sl(n+1,ℂ)$.

Let $R$ be an associative ring with nonzero identity. The zero-divisor graph $\mathrm{\Gamma }(R)$ of $R$ is the (undirected) graph with vertices the nonzero zero-divisors of $R$, and distinct vertices $a$ and $b$ are adjacent if and only if $ab=0$ or $ba=0$. Let $r_R(a)$ and $l_R(a)$ be the set of all right annihilators and the set of all left annihilator of an element $a\in R$, respectively, and let ${\mathrm{\text{ann}}}_R(a)=l_R(a)\cup r_R(a)$. The relation on $R$ given by $a\sim b$ if and only if ${\mathrm{\text{ann}}}_R(a)={\mathrm{\text{ann}}}_R(b)$ is an equivalence relation. The compressed zero-divisor graph ${\mathrm{\Gamma }}_E(R)$ of $R$ is the (undirected) graph with vertices the equivalence classes induced by $\sim$ other than ${[0]}_R$ and ${[1]}_R$, and distinct vertices ${[a]}_R$ and ${[b]}_R$ are adjacent if and only if $ab=0$ or $ba=0$. The goal of our paper is to study the diameter of zero-divisor and the compressed zero-divisor graph of skew Laurent polynomial rings over noncommutative rings. We give a complete characterization of the possible diameters of $\mathrm{\Gamma }(R[x,x^{-1};\alpha ])$ and ${\mathrm{\Gamma }}_E(R[x,x^{-1};\alpha ])$, where the base ring $R$ is reversible and also has the $\alpha$-compatible property.

We investigate conditions when the $t$-structure of Happel–Reiten–Smalø associated to a torsion pair is a compactly generated $t$-structure. The concept of a $t$CG torsion pair is introduced and for any ring $R$, we prove that $t=({𝒯},{ℱ})$ is a $t$CG torsion pair in $R\mathrm{\text{-Mod}}$ if, and only if, there exists, $\{T_{\lambda }\}$ a set of finitely presented $R$-modules in ${𝒯}$, such that ${ℱ}=\bigcap \mathrm{\text{Ker}}({\mathrm{\text{Hom}}}_R(T_{\lambda },?))$. We also show that every $t$CG torsion pair is of finite type, and show that the reciprocal is not true. Finally, we give a precise description of the $t$CG torsion pairs over Noetherian rings and von Neumman regular rings.

In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements $a,b$ of a von Neumann regular ring $R$, $a=b$ if and only if $I(a)=I(b)$, where $I(x)$ denotes the set of inner inverses of $x\in R$. We also prove that, in a semiprime ring, the same is true for reflexive inverses.

Let $R$ be an arbitrary ring and $M$ be a right $R$-module. A proper submodule $N$ of $M$ is called almost prime (respectively, weakly prime) if for each submodule $K$ of $M$ and each ideal $I$ of $R$ that $KI\subseteq N$ and $KI⊈N(N{:}_{R}M)$ (respectively, $KI\ne 0$), then $K\subseteq N$ or $I\subseteq (N{:}_{R}M)$. We study these notions which are new generalizations of the prime submodules over noncommutative rings and we obtain some related results. We show that these two concepts in some classes of modules coincide. Moreover, we investigate the conditions that $NI$ is almost prime, where $N$ is a submodule of $M$ and $I$ is an ideal of $R$. Also, the almost prime radical of modules will be introduced and we extend some known results.

Given a representation $(\rho ;A)$ of a 3-Lie algebra $B$, we construct first-order cohomology classes by using derivations of $A$, $B$ and obtain a Lie algebra $G_{\rho }$ with a representation $\mathrm{\Phi }$ on $H^1(B;A)$. In the case that $\rho$ is given by an abelian extension $0\to A\hookrightarrow L\to B\to 0$ of 3-Lie algebras with $[A,A,L]=0$, we obtain obstruction classes for extensibility of derivations of $A$ and $B$ to those of $L$. An application of the representation $\mathrm{\Phi }$ to derivations is also discussed.

We study Zariski-like topologies on a proper class $X⫋L$ of a complete lattice $\mathcal{ℒ}=(L,\wedge ,\vee ,0,1)$. We consider $X$ with the so-called classical Zariski topology $(X,{\tau }^{cl})$ and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that $\mathcal{ℒ}$ is $X$-top if

Let $K/F$ be a quadratic tamely ramified extension of a non-Archimedean local field $F$ of characteristic zero. In this paper, we give an explicit formula for Langlands’ lambda function ${\lambda }_{K/F}$.

An element $a$ in a ring $R$ is called clean if it is the sum of an idempotent $e$ and a unit $u$. Such a clean decomposition $a=e+u$ is said to be strongly clean if $eu=ue$ and special clean if $aR\cap eR=(0)$. In this paper, we prove that $a$ is Drazin invertible if and only if there exists an idempotent $e$ and a unit $u$ such that $a^n=e+u$ is both a strongly clean decomposition and a special clean decomposition, for some positive integer $n$. Also, the existence of the Moore–Penrose and group inverses is related to the existence of certain ∗-clean decompositions.

Let $N(𝔽)$ be the Lie algebra consisting of all strictly upper triangular $(n+1)\times (n+1)$ matrices over a field $𝔽$. An invertible linear map $\phi$ on $N(𝔽)$ is called to be strong commutativity preserving (simply denoted by SCP) if $[\phi (x),\phi (y)]=[x,y]$ for any $x,y\in N(𝔽)$. We show that for $n\ge 4$, an invertible linear map $\phi$ preserves strong commutativity if and only if there exist $b,b_1,b_2\in 𝔽$ and a linear function $f:N(𝔽)\to 𝔽$ with $f(E_{1,n+1})\ne -1$ such that $\phi =\pm {\sigma }_b{\mu }_{b_2}^{(n,n+1)}{\mu }_{b_1}^{(12)}{\phi }_f$, where ${\sigma }_b$ is an inner SCP, ${\mu }_{b_2}^{(n,n+1)},{\mu }_{b_1}^{(12)}$ are extremal SCPs, ${\phi }_f$ is a central SCP.

Given a genus 2 curve $\mathrm{\text{C}}$ defined over a finite field ${𝔽}_q$ of odd characteristic such that $2|\#\mathrm{\text{Jac}}(\mathrm{\text{C}})({𝔽}_q)$, we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of ${𝔽}_q$. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of $\mathrm{\text{Jac}}(\mathrm{\text{C}})({𝔽}_{q^{2^k}})$.

We provide a diagrammatic approach to compute the algebraic structures on the Grothendieck group of a tower of the Temperley–Lieb algebras.

Given an $R$-module $C$ and a class of $R$-modules ${𝒟}$ over a commutative ring $R$, we investigate the relationship between the existence of ${𝒟}$-envelopes (respectively, ${𝒟}$-covers) and the existence of $\mathrm{\text{Hom}}(C,{𝒟})$-envelopes or $C\otimes {𝒟}$-envelopes (respectively, $\mathrm{\text{Hom}}(C,{𝒟})$-covers or $C\otimes {𝒟}$-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect rings and Artinian rings in terms of envelopes and covers by $C$-projective, $C$-flat, $C$-injective and $C$-$FP$-injective modules, where $C$ is a semidualizing $R$-module.

In a 2002 paper, P. Goeters and B. Olberding compare local, near, and stable isomorphisms of torsionless modules over $h$-local domains. In this paper, we compare these weaker forms of isomorphisms of torsionless modules over domains of finite character.

We provide a new approach to the investigation of maximal commutative subalgebras (with respect to inclusion) of Grassmann algebras. We show that finding a maximal commutative subalgebra in Grassmann algebras is equivalent to constructing an intersecting family of subsets of various odd sizes in $[n]$ which satisfies certain combinatorial conditions. Then we find new maximal commutative subalgebras in the Grassmann algebra of odd rank $n$ by constructing such combinatorial systems for odd $n$. These constructions provide counterexamples to conjectures made by Domoskos and Zubor.

Let ${𝒵}$ be a specialization closed subset of Spec R and X a homologically left-bounded complex with finitely generated homologies. We establish Faltings’ Local-global Principle and Annihilator Theorems for the local cohomology modules ${\mathrm{\text{H}}}_{{𝒵}}^i(X).$ Our versions contain variations of results already known on these theorems.