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...

Let $R$ be a ring. A subclass $𝒯$ of left $R$-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. Then a left $R$-module $M$ is called $𝒯$-finitely generated if there exists a finitely generated submodule $N$ such that $M/N\in 𝒯$; a left $R$-module $A$ is called $(𝒯,n)$-presented if there exists an exact sequence of left $R$-modules $$0⟶K_{n-1}⟶F_{n-1}⟶\cdots ⟶F_1⟶F_0⟶M⟶0$$ such that $F_0,\cdots ,F_{n-1}$ are finitely generated free and $K_{n-1}$ is $𝒯$-finitely generated; a left $R$-module...

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.

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...

First, we give complete description of the comultiplication modules over a Dedekind domain. Second, if $R$ is the pullback of two local Dedekind domains, then we classify all indecomposable comultiplication $R$-modules and establish a connection between the comultiplication modules and the pure-injective modules over such domains.

In this paper, we show the existence of copure injective preenvelopes over noetherian rings and copure flat preenvelopes over commutative artinian rings. We use this to characterize $n$-Gorenstein rings. As a consequence, if the full subcategory of strongly copure injective (respectively flat) modules over a left and right noetherian ring $R$ has cokernels (respectively kernels), then $R$ is $2$-Gorenstein.