arXiv subjects

Antonio de França

Publications and source records attributed to Antonio de França.

5 recordsLinked to original sources

arXiv2025

The Graded Algebras with a Graded Identity of Degree 2

This paper is devoted to the study of graded associative algebras that satisfy a graded polynomial identity of degree $2$. % Let $\mathsf{G}$ be a finite abelian group, $\mathbb{F}$ a field of characteristic zero and $\mathfrak{A}$ a $\mathsf{G}$-graded $\mathbb{F}$-algebra. % We prove that, for $\mathbb{F}$ algebraically closed, if $\mathfrak{A}_e$ satisfies a polynomial identity $g=g(x_1^{(e)}, \dots, x_n^{(e)})\in\mathbb{F}\langle X^\mathsf{G} \rangle$ of degree $2$, then $\mathfrak{A}$ is either nilpotent or has commutative neutral component, % and we ensure that the $\mathsf{G}$-graded variety $\mathfrak{W}^\mathsf{G}$ determined by $g$ is equal to either $\mathsf{var}^\mathsf{G}([x^{(e)},y^{(e)}])$ or $\mathsf{var}^\mathsf{G}(N)$ for some nilpotent $\mathsf{G}$-graded algebra $N$. % Posteriorly, we investigate the implications of $\mathfrak{A}_e$ being central in $\mathfrak{A}$. The results obtained allow us to prove that, when $\mathsf{G}$ is finite cyclic, if $\mathfrak{A}$ is finitely generated and $\mathfrak{A}_e$ is central in $\mathfrak{A}$, then the commutator ideal of $\mathfrak{A}$ is nilpotent, and the algebra $\mathfrak{A}^{(-)}=(\mathfrak{A},[\ ,\ ])$ is a solvable Lie algebra, % and, if $\mathsf{G}$ has odd order, then $[x_1,x_2][x_3,x_4]\cdots[x_{2d-1},x_{2d}]\equiv0$ in $\mathfrak{A}$, for some $d\in\mathbb{N}$.

math.RA↗

arXiv2025

The Nilpotency of the Nil Metric $\mathbb{F}$-Algebras

Let $\mathbb{F}$ be a normed field. In this work, we prove that every nil complete metric $\mathbb{F}$-algebra is nilpotent when $\mathbb{F}$ has characteristic zero. This result generalizes Grabiner's Theorem for Banach algebras, first proved in 1969. Furthermore, we show that a metric $\mathbb{F}$-algebra $\mathfrak{A}$ and its completion $C(\mathfrak{A})$ satisfy the same polynomial identities, and consequently, if $\mathsf{char}(\mathbb{F})=0$ and $C(\mathfrak{A})$ is nil, then $\mathfrak{A}$ is nilpotent. Our results allow us to resolve Köthe's Problem affirmatively for complete metric algebras over normed fields of characteristic zero.

math.RA↗

arXiv2024

Graded Imbeddings in Finite Dimensional Simple Graded Algebras

Let $\mathbb{F}$ be a field and $\mathsf{G}$ a group. This work is inspired in the following problem: "{\it given a division (simple) $\mathsf{G}$-graded $\mathbb{F}$-algebra, is there any other division (simple) $\mathsf{G}$-graded $\mathbb{F}$-algebra such that the former can be $\mathsf{G}$-imbedded in the latter?}". In this work, we answer this question affirmatively for $\mathbb{F}$ algebraically closed, $\mathsf{G}$ finite abelian, and associative algebras of finite dimension. To prove this, we apply concepts and properties of Group Cohomology. We show $\mathcal{H}^2(H, \mathbb{F}^*)=\mathsf{res}^\mathsf{G}_H \left(\mathcal{H}^2(\mathsf{G}, \mathbb{F}^*)\right)$, where $H$ is a subgroup of $\mathsf{G}$ and $\mathsf{res}^\mathsf{G}_H$ is the restriction homomorphism. Posteriorly, we prove that, given any $H_1,H_2\leq\mathsf{G}$ and $σ_i\in\mathcal{Z}^2(H_i,\mathbb{F}^*)$, $i=1,2$, are equivalent: i) $\mathbb{F}^{σ_1}[H_1] \stackrel{\mathsf{G}}{\hookrightarrow} \mathbb{F}^{σ_2}[H_2]$; ii) $H_1\leq H_2$ and $[σ_1]=[σ_2]_{H_1}$; iii) $\mathsf{T}^{\mathsf{G}}(\mathbb{F}^{σ_2}[H_2])\subseteq \mathsf{T}^{\mathsf{G}}(\mathbb{F}^{σ_1}[H_1])$, where $\mathsf{T}^{\mathsf{G}}(\mathbb{F}^{σ_i}[H_i])$ is the $\mathsf{G}$T-ideal of graded identities of $\mathbb{F}^{σ_i}[H_i]$. Furthermore, we prove that, given $\mathfrak{A}$ and $\mathfrak{B}$ two finite dimensional simple $\mathsf{G}$-graded $\mathbb{F}$-algebras, if $\mathbb{F}$ is algebraically closed, $\mathsf{char}(\mathbb{F}) = 0$ or $\mathsf{char} (\mathbb{F})$ is coprime with the order of each finite subgroup of $\mathsf{G}$, and any subgroup of $\mathsf{G}$ is normal, then $\mathsf{T}^{\mathsf{G}}(\mathfrak{A})\subseteq \mathsf{T}^{\mathsf{G}}(\mathfrak{B})$ iff $\mathfrak{B} \stackrel{\mathsf{G}}{\hookrightarrow} \mathfrak{A}$.

math.RA↗

arXiv2024

Irreducible Graded Bimodules over Algebras and a Pierce Decomposition of the Jacobson Radical

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let $\mathsf{G}$ be a group, $\mathbb{F}$ an algebraically closed field with $\mathsf{char}(\mathbb{F})=0$, $\mathfrak{A}$ a finite dimensional $\mathsf{G}$-graded associative $\mathbb{F}$-algebra and $\mathsf{M}$ a $\mathsf{G}$-graded unitary $\mathfrak{A}$-bimodule. We proved that if $\mathfrak{A}=M_n(\mathbb{F}^σ[\mathsf{H}])$ with a canonical elementary $\mathsf{G}$-grading, where $\mathsf{H}$ is a finite abelian subgroup of $\mathsf{G}$ and $σ\in\mathsf{Z}^2(\mathsf{H},\mathbb{F}^*)$, then $\mathsf{M}$ being irreducible graded implies that there exists a nonzero homogeneous element $w\in\mathsf{M}$ satisfying $\mathsf{M}=\mathfrak{B}w$ and $\mathfrak{B} w= w\mathfrak{B}$. Another result we proved generalizes the last one: if $\mathsf{G}$ is abelian, $\mathfrak{A}$ is simple graded and $\mathsf{M}$ is finitely generated, then there exist nonzero homogeneous elements $w_1, w_2,\dots,w_n\in\mathsf{M}$ such that \begin{equation}\nonumber \mathsf{M}=\mathfrak{A} w_1\oplus\mathfrak{A} w_2\oplus \cdots \oplus \mathfrak{A} w_n \ , \end{equation} where $w_i \mathfrak{A}=\mathfrak{A} w_i\neq0$ for all $i=1, 2,\dots,n$, and each $\mathfrak{A} w_i$ is irreducible. The elements $w_i$'s are associated with the irreducible characters of $\mathsf{G}$. We also describe graded bimodules over graded semisimple algebras. And we finish by presenting a Pierce decomposition of the graded Jacobson radical of any finite dimensional $\mathbb{F}$-algebra with a $\mathsf{G}$-grading.

math.RT↗

arXiv2021

Köthe's Problem, Kurosch-Levitzki Problem and Graded Rings

Let $\mathfrak{R}$ be an associative ring graded by left cancellative monoid $\mathsf{S}$, and $e$ the neutral element of $\mathsf{S}$. We study the following problem: if $\mathfrak{R}_e$ is nil, then is $\mathfrak{R}$ nil/nilpotent? We have proved that if $\mathfrak{R}_e$ is nil (of bounded index) and $\mathsf{f}$- commutative, then $\mathfrak{R}$ is nil (of bounded index). Later, we have shown that $\mathfrak{R}_e$ being nilpotent implies $\mathfrak{R}$ is nilpotent. Consequently, we have exhibited a generalization of Dubnov-Ivanov-Nagata-Higman Theorem for the graded algebras case. Furthermore, we have exhibited relations between graded rings and the problems of Köthe and Kurosh-Levitzki. We have proved that graded rings and $\mathsf{f}$-commutative rings provide positive solutions to these problems.

math.RA↗