arXiv subjects

Edward Landi Tonucci

Publications and source records attributed to Edward Landi Tonucci.

3 recordsLinked to original sources

arXiv2025

On $*$-Clean Group Rings over SLC-groups

The property of $*$-cleanness in group rings has been studied for some groups considering the classical involution, given by $g^*=g^{-1}$. A group is called an SLC-group if its quotient by its center is isomorphic to the Klein group; these groups are equipped with its own canonical involution, which usually does not coincide with the classical one. In this paper we study the $*$-cleanness of $RG$ when $G$ is an SLC-group, considering $*$ as its canonical involution. In that context, we prove that if $RG$ is $*$-clean then $G$ is the direct product of $Q_8$ and an abelian group with some extra properties and we find a converse for some specific cases, generalizing a result by Gao, Chen and Li for $Q_8$.

math.RA↗

arXiv2015

Anticommutativity of Skew-symmetric Elements under Generalized Oriented Involutions

Let $R$ be a ring with $char(R)\neq2$ whose unit group are denoted by $\mathcal{U}(R)$, $G$ a group, and $RG$ its group ring. Let $*$ be an involution in $G$, $σ:G\rightarrow\mathcal{U}(R)$ be a nontrivial group homomorphism, with $ker\ σ=N$, satisfying $xx^*\in N$ for all $x\in G$, and define the generalized oriented involution $σ*$ in $RG$ by $\left( \sum_{x\in G}α_xx\right)^{σ*}=\sum_{x\in G}σ(x)α_xx^*$. An element $α\in RG$ is called skew-symmetric if $α^{σ*}=-α$, and the set of all skew-symmetric elements are denoted by $(RG)^-$. In this paper, we will classify the group rings $RG$ such that $(RG)^-$ is anticommutative, generalizing, and obtaining as consequence, the main result of \cite{GP13a}.

math.RA↗

arXiv2015

Anticommutativity of Symmetric Elements under Generalized Oriented Involutions

Let $R$ be a ring with $char(R)\neq2$ whose unit group are denoted by $\mathcal{U}(R)$, $G$ a group with involution $*$, and $σ:G\rightarrow\mathcal{U}(R)$ a nontrivial group homomorphism, with $ker\ σ=N$, satisfying $xx^*\in N$ for all $x\in G$. Let $RG$ be the group ring of $G$ over $R$ and define the involution $σ*$ in $RG$ by $\left( \sum_{x\in G}α_xx\right)^{σ*}=\sum_{x\in G}σ(x)α_xx^*$. In this paper, we will classify the group rings $RG$ such that $\mathcal{S}$ is anticommutative, where $\mathcal{S}$ is the largest subset of $(RG)^+=\left\{ α\in RG: α^{σ*}=α\right\}$ that can satisfy anticommutativity under $char(R)\neq2$.

math.RA↗