Let $m\ge 0,r\ge 0,s\ge 0,q\ge 0$ be fixed integers. Suppose that $R$ is an associative ring with unity $1$ in which for each $x,y\in R$ there exist polynomials $f\left(X\right)\in X^{2}ZZ\left[X\right],g\left(X\right),h\left(X\right)\in XZZ\left[X\right]$ such that $\{1-g\left(y{x}^m\right)\}[x,x^{r}y-x^{s}f\left(y{x}^m\right)x^q]\{1-h\left(y{x}^m\right)\}=0$. Then $R$ is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of $x$ and $y$. Finally, commutativity of one sided s-unital ring is also obtained when $R$ satisfies some related ring properties.

In this paper we investigate commutativity of rings with unity satisfying any one of the properties: $$\begin{array}{cc}& \{1-g\left(y{x}^m\right)\}[y{x}^m-x^{r}f\left(y{x}^m\right)x^s,x]\{1-h\left(y{x}^m\right)\}=0,\hfill \\ & \{1-g\left(y{x}^m\right)\}[x^{m}y-x^{r}f\left(y{x}^m\right)x^s,x]\{1-h\left(y{x}^m\right)\}=0,\hfill \\ & y^t[x,y^n]=g\left(x\right)[f\left(x\right),y]h\left(x\right)\mathrm{a}nd[x,y^n]y^t=g\left(x\right)[f\left(x\right),y]h\left(x\right)\hfill \end{array}$$ for some $f\left(X\right)$ in $X^2\mathbb{Z}\left[X\right]$ and $g\left(X\right)$, $h\left(X\right)$ in $\mathbb{Z}\left[X\right]$, where $m\ge 0$, $r\ge 0$, $s\ge 0$, $n>0$, $t>0$ are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements $x$ and $y$ for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize...

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...