### ${\mathbb{Z}}_{k+l}\times {\mathbb{Z}}_{2}$-graded polynomial identities for ${M}_{k,l}\left(E\right)\otimes E$

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2000 Mathematics Subject Classification: 16R10, 16R20, 16R50The algebra Mn(K) of the matrices n × n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities: x = 0, |α(x)| ≥ n, xy = yx, α(x) = α(y) = 0, xyz = zyx, α(x) = −α(y) = α(z ), where α is the degree of the corresponding variable. This is a generalization of a result of Vasilovsky about the Z-graded identities...

Let $m>1,s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p=p\left(x\right)\ge 0,q=q\left(x\right)\ge 0,n=n\left(x\right)\ge 0,r=r\left(x\right)\ge 0$ such that either ${x}^{p}[{x}^{n},y]{x}^{q}={x}^{r}[x,{y}^{m}]{y}^{s}$ or ${x}^{p}[{x}^{n},y]{x}^{q}={y}^{s}[x,{y}^{m}]{x}^{r}$ for all $y\in R$. In the present paper it is shown that $R$ is commutative if it satisfies the property $Q\left(m\right)$ (i.e. for all $x,y\in R,m[x,y]=0$ implies $[x,y]=0$).

The purpose of this paper is to prove the following result: Let $R$ be a $2$-torsion free semiprime ring and let $T:R\to R$ be an additive mapping, such that $2T\left({x}^{2}\right)=T\left(x\right)x+xT\left(x\right)$ holds for all $x\in R$. In this case $T$ is left and right centralizer.

Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z\left(U\right)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $\left[F\right(u),u]F\left(u\right)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F\left(x\right)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity ${s}_{4}$ and there exist $a\in U$ and $\alpha \in C$ such that $F\left(x\right)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of...

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 $D$ be a division ring finite dimensional over its center $F$. The goal of this paper is to prove that for any positive integer $n$ there exists $a\in {D}^{\left(n\right)},$ the $n$th multiplicative derived subgroup such that $F\left(a\right)$ is a maximal subfield of $D$. We also show that a single depth-$n$ iterated additive commutator would generate a maximal subfield of $D.$

Let $R$ be an associative ring with identity $1$ and $J\left(R\right)$ the Jacobson radical of $R$. Suppose that $m\ge 1$ is a fixed positive integer and $R$ an $m$-torsion-free ring with $1$. In the present paper, it is shown 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) $[x,[x,{y}^{m}]]=0$, for all $x,y\in R\setminus J\left(R\right)$. This result is also valid if (ii) is replaced by (ii)’ $[{\left(yx\right)}^{m}{x}^{m}-{x}^{m}{\left(xy\right)}^{m},x]=0$, for all $x,y\in R\setminus N\left(R\right)$. Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).

In this paper we investigate commutativity of rings with unity satisfying any one of the properties: $$\begin{array}{cc}& \{1-g\left(y{x}^{m}\right)\}\phantom{\rule{4pt}{0ex}}[y{x}^{m}-{x}^{r}f\left(y{x}^{m}\right)\phantom{\rule{4pt}{0ex}}{x}^{s},x]\{1-h\left(y{x}^{m}\right)\}=0,\hfill \\ & \{1-g\left(y{x}^{m}\right)\}\phantom{\rule{4pt}{0ex}}[{x}^{m}y-{x}^{r}f\left(y{x}^{m}\right){x}^{s},x]\{1-h\left(y{x}^{m}\right)\}=0,\hfill \\ & {y}^{t}[x,{y}^{n}]=g\left(x\right)[f\left(x\right),y]h\left(x\right)\phantom{\rule{4pt}{0ex}}\mathrm{a}nd\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}[x,{y}^{n}]\phantom{\rule{4pt}{0ex}}{y}^{t}=g\left(x\right)[f\left(x\right),y]h\left(x\right)\hfill \end{array}$$ 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{'}}$$...$