Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products $(A{♮}_{R}B,\alpha \otimes \beta )$. Moreover, necessary and sufficient conditions for $(A{♮}_{R}B,\alpha \otimes \beta )$ to be a cobraided Hom-Hopf algebra are given.

Soit $A_1\left(ℂ\right)$ la première algèbre de Weyl sur $ℂ$. La codimension B-W d’un idéal à droite non nul $I$ de $A_1\left(ℂ\right)$ a été introduite par Yuri Berest et George Wilson. Nous montrons d’une part que cette codimension est invariante par la relation de Stafford : si $x\in Q_1=\mathrm{Frac}\left(A_1\left(ℂ\right)\right)$, le corps de fractions de $A_1\left(ℂ\right)$, et si $\sigma \in \mathrm{Aut}\left(A_1\left(ℂ\right)\right)$, le groupe des $ℂ$-automorphismes de $A_1\left(ℂ\right)$, sont tels que $J=x\sigma \left(I\right)$ soit un idéal à droite de $A_1\left(ℂ\right)$, alors $\mathrm{codim}I=\mathrm{codim}x\sigma \left(I\right)$. Nous relions d’autre part la codimension d’un idéal $I$ à la codimension de Gail Letzter-Makar Limanov, de $\mathrm{End}\left(I\right)$, l’anneau des endomorphismes...

We give axiomatic conditions in order to calculate the local cohomology of some idempotent kernel functors. These results lie in some new dimension introduced by T. Levasseur for Auslander-Gorenstein rings. Under some hypothesis, we generalize previous results.

Let $G$ be a finite group with a normal subgroup $N$ such that $C_G\left(N\right)\le N$. It is shown that under some conditions, Coleman automorphisms of $G$ are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.

A new class of abelian $p$-groups with all high subgroups isomorphic is defined. Commutative modular and semisimple group algebras over such groups are examined. The results obtained continue our recent statements published in Comment. Math. Univ. Carolinae (2002).

Let $G$ be a $p$-mixed abelian group and $R$ is a commutative perfect integral domain of $charR=p>0$. Then, the first main result is that the group of all normalized invertible elements $V\left(RG\right)$ is a $\Sigma$-group if and only if $G$ is a $\Sigma$-group. In particular, the second central result is that if $G$ is a $\Sigma$-group, the $R$-algebras isomorphism $RA\cong RG$ between the group algebras $RA$ and $RG$ for an arbitrary but fixed group $A$ implies $A$ is a $p$-mixed abelian $\Sigma$-group and even more that the high subgroups of $A$ and $G$ are isomorphic, namely, ${\mathscr{H}}_A\cong {\mathscr{H}}_G$. Besides,...

We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra $s{l}_2$, the Verma module over a Kac-Moody algebra, the Verma module...