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In this article we present set of infinite natural numbers which satisfies the conjecture $3n+1$., Comment: The article is written in format of a report

A conjecture due to Zassenhaus asserts that if $\ G$ is a finite group then any torsion unit in $\mathbb{Z}G$ is conjugate in $\mathbb{Q}G$ to an element of $\ G$. We present a weaker form of this conjecture for some infinite groups., Comment: Key words and frases: Group Rings, torsion units, unique trace property, 7 pages

In \cite{jpsf} we constructed pairs of units $u,v$ in $\Z$-orders of a quaternion algebra over $\Q (\sqrt{-d})$, $d \equiv 7 \pmod 8$ positive and square free, such that $< u^ n,v^n>$ is free for some $n\in \mathbb{N}$. Here we extend this result to any imaginary quadratic extension of $\ \mathbb{Q}$, thus including matrix algebras. More precisely, we show that $< u^n,v^n> $ is a free group for all $n\geq 1$ and $d>2$ and for $d=2$ and all $n\geq 2$. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also established and used to prove that two certain units generate a free semigroup but that, in this case, the Ping-Pong Lemma can not be applied to show that the group they generate is free., Comment: 10 pages, article presented in conferences: Algebra School, Brasilia-Brazil, Brasilia National University (july-2010); Summer 2009 Meeting of CMS in Groups and Hopf Algebras section, St. Jonh's-Canada, Memorial University of Newfoundland (June-2009); Groups, Rings and Group Rings, Ubatuba-Brazil (july-2008)

In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra %over a field of characteristic zero, contains a ${\mathbb{Z}}$-order with hyperbolic unit group. In this paper we complete this classification by handling the case in which the semigroup is semi-simple., Comment: In doi:10.1016/j.jalgebra.2008.03.015 it was, partially, classified the finite semigroups S which the semigroup algebra QS has the hyperbolic property. In this article we complete this classification

We investigate the structure of an alternative finite dimensional $\Q$-algebra $\mathfrak{A}$ subject to the condition that for a $\Z$-order $\Gamma \subset \mathfrak{A}$, and thus for every $\Z$-order of $\mathfrak{A}$, the loop of units of $\U (\Gamma)$ does not contain a free abelian subgroup of rank two. In particular, we prove that the radical of such an algebra associates with the whole algebra. We also classify $RA$-loops $L$ for which $\mathbb{Z}L$ has this property. The classification for group rings is still an open problem., Comment: Third author's Ph.D. (parcial); generalization of those results for the non-associative case, 9 pps. Conf.: Geometry, Topology, Algebra and Number Theory, Steklov Math. Inst., Moscow-Russia (Aug, 2010); XVIII Latin American Colloquium of Algebra, S\~ao Paulo-Brazil (July, 2009); Algebras, Representation and Applications, Ubatuba-Brazil (aug., 2007); Group and Group Rings, Bedlewo-Poland (2005)

We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct units in a non-split quaternion algebra over R., Comment: 15 pages, this work is part of the PHd. Thesis of the third author. The paper was accepted in Proceedings Mathematical Sciences

Let $A$ be a finite dimensional $Q-$algebra and $\Gamma subset A$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\mathcal{U}(\Gamma)$. We call this last property the hyperbolic property. We apply this in the case that $A = KS$ a semigroup algebra with $K = Q$ or $K = Q(\sqrt{-d})$. In particular, when $KS$ is semi-simple and has no nilpotent elements, we prove that $S$ is an inverse semigroup which is the disjoint union of Higman groups and at most one cyclic group $C_n$ with $n \in \{5,8,12\}$., Comment: This article corresponds to the second chapter of the third author PhD Thesis

We classify the finite semigroups S, for which all the Z-orders O of the rational Q-algebra QS, is such that the unit group U(O) is hyperbolic. We also classify the RA-loops L, for which the unit loop U(ZL) does not contain any free abelian subgroup of rank two., Comment: 5 pages; this is part of the second and third chapters of the second author's PhD. Thesis. The last theorem was revised, as well as added some definitions and references