The main result: Let $R$ be a $2$-torsion free semiprime ring and let $T:R\to R$ be an additive mapping. Suppose that $T\left(xyx\right)=xT\left(y\right)x$ holds for all $x,y\in R$. In this case $T$ is a centralizer.

Let $ℛ$ be a commutative ring, $𝒢$ be a generalized matrix algebra over $ℛ$ with weakly loyal bimodule and $𝒵\left(𝒢\right)$ be the center of $𝒢$. Suppose that $𝔮:𝒢\times 𝒢\to 𝒢$ is an $ℛ$-bilinear mapping and that ${𝔗}_{𝔮}:𝒢\to 𝒢$ is a trace of $𝔮$. The aim of this article is to describe the form of ${𝔗}_{𝔮}$ satisfying the centralizing condition $[{𝔗}_{𝔮}\left(x\right),x]\in 𝒵\left(𝒢\right)$ (and commuting condition $[{𝔗}_{𝔮}\left(x\right),x]=0$) for all $x\in 𝒢$. More precisely, we will revisit the question of when the centralizing trace (and commuting trace) ${𝔗}_{𝔮}$ has the so-called proper form from a new perspective. Using the aforementioned...

Let $(𝔄,{ϵ}_u)$ and $(𝔅,{ϵ}_b)$ be two pointed sets. Given a family of three maps $ℱ=\{f_1:𝔄\to 𝔄;f_2:𝔄\times 𝔄\to 𝔄;f_3:𝔄\times 𝔄\to 𝔅\}$, this family provides an adequate decomposition of $𝔄\setminus \left\{{ϵ}_u\right\}$ as the orthogonal disjoint union of well-described $ℱ$-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak $H^{*}$-algebras.

We give some sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, normal EP element and strongly EP element by using solutions of certain equations.

Si considerano le estensioni chiuse $B$ di un $R$-modulo $A$ mediante un $R$-modulo $C$ nel caso in cui $R$ sia un anello semi-artiniano, cioè un anello $R$ con la proprietà che per ogni quoziente $(R{/}_I)\ne 0$ sia soc $(R/I)\ne 0$. Tali estensioni sono caratterizzate dal fatto che $A$ deve essere un sottomodulo semi-puro di $B$.

We develop a technique for the study of K-coalgebras and their representation types by applying a quiver technique and topologically pseudocompact modules over pseudocompact K-algebras in the sense of Gabriel [17], [19]. A definition of tame comodule type and wild comodule type for K-coalgebras over an algebraically closed field K is introduced. Tame and wild coalgebras are studied by means of their finite-dimensional subcoalgebras. A weak version of the tame-wild dichotomy theorem of Drozd [13]...

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...

For a split graph order ℒ over a complete local regular domain $𝒪$ of dimension 2 the indecomposable Cohen-Macaulay modules decompose - up to irreducible projectives - into a union of the indecomposable Cohen-Macaulay modules over graph orders of type •—• . There, the Cohen-Macaulay modules filtered by irreducible Cohen-Macaulay modules are in bijection to the homomorphisms $\varphi :𝒪L^{\left(\mu \right)}\to 𝒪L^{\left(\nu \right)}$ under the bi-action of the groups $\left(Gl\right(\mu ,𝒪L),Gl(\nu ,𝒪L\left)\right)$, where $𝒪L=𝒪/〈\pi 〉$ for a prime π. This problem strongly depends on the nature of $𝒪L$. If $𝒪L$ is regular,...