arXiv subjects

Mohamed Louzari

Publications and source records attributed to Mohamed Louzari.

10 recordsLinked to original sources

arXiv2026

Kählerian oxidation and reduction of Lie algebras with Kähler structures

A Kähler structure on a real Lie algebra $\mathfrak{g}$ is a pair $(ω, J)$ of a 2-cocycle and a complex structure on $\mathfrak{g}$ which are compatible in the sense that $J$ is skew-symmetric with respect to $ω$. This paper investigates Kähler structures on Lie algebras, with a particular focus on the nilpotent case. We introduce an algebraic framework for the Kählerian reduction and oxidation of Lie algebras by one-dimensional and two-dimensional ideals. Our main structural result demonstrates that every indecomposable, quasi-nilpotent Kählerian nilpotent Lie algebra of dimension greater than four can be inductively reconstructed through a finite sequence of Kählerian central oxidations by planes. This reduction sequence originates from a base algebra that is either $\{0\}$, $\mathbb{R}^2$, or one admitting a strongly non-nilpotent complex structure. Moreover, if the complex structure is nilpotent, all intermediate reductions also have a nilpotent complex structure.

math.DG↗

arXiv2023

Generalized rigid modules and their polynomial extensions

Let $R$ be a ring with unity, $σ$ an endomorphism of $R$ and $M_R$ a right $R$-module. In this paper, we continue studding $σ$-rigid modules that were introduced by Gunner et al. \cite{generalized/rigid}. We give some results on $σ$-rigid modules and related concepts. Also, we study the transfer of $σ$-rigidness from a module $M_R$ to its extensions, as triangular matrix modules and polynomial modules.

math.RA↗

arXiv2023

On pseudo-Hermitian quadratic nilpotent Lie algebras

We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.

math.RA↗

arXiv2017

Skew polynomial rings over abelian and idempotent reflexive rings

Let $R$ be a ring and $σ$ an endomorphism of $R$. In this note, we study skew polynomial rings and skew power series rings over idempotent reflexive rings and abelian rings. Also, we introduce the concept of right (resp., left) $σ$-idempotent reflexive rings which generalizes right (resp., left) idempotent reflexive rings and $σ$-abelian rings. Certain results are obtained as corollaries from our results.

math.RA↗

arXiv2017

On $(σ,δ)$-skew McCoy modules

Let $(σ,δ)$ be a quasi derivation of a ring $R$ and $M_R$ a right $R$-module. In this paper, we introduce the notion of $(σ,δ)$-skew McCoy modules which extends the notion of McCoy modules and $σ$-skew McCoy modules. This concept can be regarded also as a generalization of $(σ,δ)$-skew Armendariz modules. Some properties of this concept are established and some connections between $(σ,δ)$-skew McCoyness and $(σ,δ)$-compatible reduced modules are examined. Also, we study the property $(σ,δ)$-skew McCoy of some skew triangular matrix extensions $V_n(M,σ)$, for any nonnegative integer $n\geq 2$. As a consequence, we obtain: (1) $M_R$ is $(σ,δ)$-skew McCoy if and only if $M[x]/M[x](x^n)$ is $(\overlineσ,\overlineδ)$-skew McCoy, and (2) $M_R$ is $σ$-skew McCoy if and only if $M[x;σ]/M[x;σ](x^n)$ is $\overlineσ$-skew McCoy.

math.RA↗

arXiv2012

On McCoy Condition and Semicommutative Rings

Let $R$ be a ring, $σ$ an endomorphism of $R$, $I$ a right ideal in $S=R[x;σ]$ and $M_R$ a right $R$-module. We give a generalization of McCoy's Theorem \cite{mccoy}, by showing that, if $r_S(I)$ is $σ$-stable or $σ$-compatible. Then $\;r_S(I)\neq 0$ implies $r_R(I)\neq 0$. As a consequence, if $R[x;σ]$ is semicommutative then $R$ is $σ$-skew McCoy. Moreover, we show that the Nagata extension $R\oplus_σM_R$ is semicommutative right McCoy when $R$ is a commutative domain.

math.RA↗

arXiv2011

On Skew Polynomials over p.q.-Baer and p.p.-Modules

Let $M_R$ be a module and $σ$ an endomorphism of $R$. Let $m\in M$ and $a\in R$, we say that $M_R$ satisfies the condition $\mathcal{C}_1$ (respectively, $\mathcal{C}_2$), if $ma=0$ implies $mσ(a)=0$ (respectively, $mσ(a)=0$ implies $ma=0$). We show that if $M_R$ is p.q.-Baer then so is $M[x;σ]_{R[x;σ]}$ whenever $M_R$ satisfies the condition $\mathcal{C}_2$, and the converse holds when $M_R$ satisfies the condition $\mathcal{C}_1$. Also, if $M_R$ satisfies $\mathcal{C}_2$ and $σ$-skew Armendariz, then $M_R$ is a p.p.-module if and only if $M[x;σ]_{R[x;σ]}$ is a p.p.-module if and only if $M[x,x^{-1};σ]_{R[x,x^{-1};σ]}$ ($σ\in Aut(R)$) is a p.p.-module. Many generalizations are obtained, and more results are found when $M_R$ is a semicommutative module.

math.RA↗

arXiv2009

A note on $σ$-reversibility and $σ$-symmetry of skew power series rings

Let $R$ be a ring and $σ$ an endomorphism of $R$. In this note, we study the transfert of the symmetry ($σ$-symmetry) and reversibility ($σ$-reversibility) from $R$ to its skew power series ring $R[[x;σ]]$. Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.p.-property of a ring $R$ and these of the skew power series ring $R[[x;σ]]$ in case $R$ is right $σ$-reversible. As a consequence we obtain a generalization of \cite{hong/2000}.

math.RA↗

arXiv2009

Ore extensions of principally quasi-Baer rings

Let $R$ be a ring and $(σ,δ)$ a quasi-derivation of $R$. In this paper, we show that if $R$ is an $(σ,δ)$-skew Armendariz ring and satisfies the condition $(\mathcal{C_σ})$, then $R$ is right p.q.-Baer if and only if the Ore extension $R[x;σ,δ]$ is right p.q.-Baer. As a consequence we obtain a generalization of \cite{hong/2000}.

math.RA↗

arXiv2009

Ore Extensions of Extended Symmetric and Reversible Rings

Let $σ$ be an endomorphism and $δ$ an $σ$-derivation of a ring $R$. In this paper, we show that if $R$ is $(σ,δ)$-skew Armendariz and $aσ(b)=0$ implies $ab=0$ for $a,b\in R$. Then $R$ is symmetric (respectively, reversible) if and only if $R$ is $σ$-symmetric (respectively, $σ$-reversible) if and only if $R[x;σ,δ]$ is symmetric (respectively, reversible). Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.q.-Baerness of a ring $R$ and these of the Ore extension $R[x;σ,δ]$. As a consequence we obtain a partial generalization of \cite{hong/2000}.

math.RA↗