Suppose that $R$ is an associative ring with identity $1$, $J\left(R\right)$ the Jacobson radical of $R$, and $N\left(R\right)$ the set of nilpotent elements of $R$. Let $m\ge 1$ be a fixed positive integer and $R$ an $m$-torsion-free ring with identity $1$. The main result of the present paper asserts that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m]=0$ for all $x,y\in R\setminus J\left(R\right)$ and (ii) $[{\left(xy\right)}^m+y^m{x}^m,x]=0=[{\left(yx\right)}^m+x^m{y}^m,x]$, for all $x,y\in R\setminus J\left(R\right)$. This result is also valid if (i) and (ii) are replaced by (i)${}^{\text{'}}$$...$

Let $p$, $q$ and $r$ be fixed non-negative integers. In this note, it is shown that if $R$ is left (right) $s$-unital ring satisfying $[f\left(x^p{y}^q\right)-x^{r}y,x]=0$ ($[f\left(x^p{y}^q\right)-y{x}^r,x]=0$, respectively) where $f\left(\lambda \right)\in {\lambda }^2\mathbb{Z}\left[\lambda \right]$, then $R$ is commutative. Moreover, commutativity of $R$ is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.

In this paper we investigate commutativity of ring $R$ with involution ${}^{\text{'}}{*}^{\text{'}}$ which admits a derivation satisfying certain algebraic identities on Jordan ideals of $R$. Some related results for prime rings are also discussed. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, $$C$$ , of a triangulated category, $$𝒯$$ , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on $$𝒯$$ whose heart is equivalent to Mod(End($$C$$ )op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave...

Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [n2/4]. In this paper, we prove this conjecture for n × n completely positive matrices over Boolean algebras (finite or infinite). In addition,we formulate various CP-rank inequalities of completely positive matrices over special semirings using semiring homomorphisms.

Nous construisons des généralisations des complexes de Koszul, associées à des symétries vérifiant l’équation de Yang-Baxter. Certains de ces complexes sont acycliques et permettent de calculer l’homologie de Hochschild et cyclique de déformations quantiques d’algèbres symétriques et extérieures. Nous donnons des résultats précis pour l’espace affine quantique multiparamétré. Il est également possible de définir des complexes de Koszul pour des algèbres enveloppantes et de Sridharan d’algèbres de...