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Certain polynomial identities and commutativity of rings

Mathematica Slovaca

Commutativity of associative rings through a Streb's classification

Archivum Mathematicum

Let $m\ge 0,\phantom{\rule{3.33333pt}{0ex}}r\ge 0,\phantom{\rule{3.33333pt}{0ex}}s\ge 0,\phantom{\rule{3.33333pt}{0ex}}q\ge 0$ be fixed integers. Suppose that $R$ is an associative ring with unity $1$ in which for each $x,y\in R$ there exist polynomials $f\left(X\right)\in {X}^{2}Z\phantom{\rule{-6.25958pt}{0ex}}Z\left[X\right],\phantom{\rule{3.33333pt}{0ex}}g\left(X\right),\phantom{\rule{3.33333pt}{0ex}}h\left(X\right)\in XZ\phantom{\rule{-6.25958pt}{0ex}}Z\left[X\right]$ such that $\left\{1-g\left(y{x}^{m}\right)\right\}\left[x,\phantom{\rule{3.33333pt}{0ex}}{x}^{r}y\phantom{\rule{3.33333pt}{0ex}}-\phantom{\rule{3.33333pt}{0ex}}{x}^{s}f\left(y{x}^{m}\right){x}^{q}\right]\left\{1-h\left(y{x}^{m}\right)\right\}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$. 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.

A commutativity theorem for associative rings

Archivum Mathematicum

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 ${x}^{p}\left[{x}^{n},y\right]{x}^{q}={x}^{r}\left[x,{y}^{m}\right]{y}^{s}$ or ${x}^{p}\left[{x}^{n},y\right]{x}^{q}={y}^{s}\left[x,{y}^{m}\right]{x}^{r}$ 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 $x,y\in R,m\left[x,y\right]=0$ implies $\left[x,y\right]=0$).

On left $\left(\theta ,\varphi \right)$-derivations of prime rings

Archivum Mathematicum

Let $R$ be a $2$-torsion free prime ring. Suppose that $\theta ,\phi$ are automorphisms of $R$. In the present paper it is established that if $R$ admits a nonzero Jordan left $\left(\theta ,\theta \right)$-derivation, then $R$ is commutative. Further, as an application of this resul it is shown that every Jordan left $\left(\theta ,\theta \right)$-derivation on $R$ is a left $\left(\theta ,\theta \right)$-derivation on $R$. Finally, in case of an arbitrary prime ring it is proved that if $R$ admits a left $\left(\theta ,\phi \right)$-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of $R$, then $d=0$...

Commutativity of *-prime rings with generalized derivations

Rendiconti del Seminario Matematico della Università di Padova

Commutativity theorems for rings with differential identities on Jordan ideals

Commentationes Mathematicae Universitatis Carolinae

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.

On Lie ideals and Jordan left derivations of prime rings

Archivum Mathematicum

Let $R$ be a 2-torsion free prime ring and let $U$ be a Lie ideal of $R$ such that ${u}^{2}\in U$ for all $u\in U$. In the present paper it is shown that if $d$ is an additive mappings of $R$ into itself satisfying $d\left({u}^{2}\right)=2ud\left(u\right)$ for all $u\in U$, then $d\left(uv\right)=ud\left(v\right)+vd\left(u\right)$ for all $u,v\in U$.

$\left(\sigma ,\tau \right)$-derivations on prime near rings

Archivum Mathematicum

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation...

On $\left(\sigma ,\tau \right)$-derivations in prime rings

Archivum Mathematicum

Let $R$ be a 2-torsion free prime ring and let $\sigma ,\tau$ be automorphisms of $R$. For any $x,y\in R$, set ${\left[x,y\right]}_{\sigma ,\tau }=x\sigma \left(y\right)-\tau \left(y\right)x$. Suppose that $d$ is a $\left(\sigma ,\tau \right)$-derivation defined on $R$. In the present paper it is shown that $\left(i\right)$ if $R$ satisfies ${\left[d\left(x\right),x\right]}_{\sigma ,\tau }=0$, then either $d=0$ or $R$ is commutative $\left(ii\right)$ if $I$ is a nonzero ideal of $R$ such that $\left[d\left(x\right),d\left(y\right)\right]=0$, for all $x,y\in I$, and $d$ commutes with both $\sigma$ and $\tau$, then either $d=0$ or $R$ is commutative. $\left(iii\right)$ if $I$ is a nonzero ideal of $R$ such that $d\left(xy\right)=d\left(yx\right)$, for all $x,y\in I$, and $d$ commutes with $\tau$, then $R$ is commutative. Finally a related result has been obtain for $\left(\sigma ,\tau \right)$-derivation....

Generalized Higher Derivations on Lie Ideals of Triangular Algebras

Communications in Mathematics

Let $𝔄=\left(\begin{array}{cc}𝒜& ℳ\\ & ℬ\end{array}\right)$ be the triangular algebra consisting of unital algebras $𝒜$ and $ℬ$ over a commutative ring $R$ with identity $1$ and $ℳ$ be a unital $\left(𝒜,ℬ\right)$-bimodule. An additive subgroup $𝔏$ of $𝔄$ is said to be a Lie ideal of $𝔄$ if $\left[𝔏,𝔄\right]\subseteq 𝔏$. A non-central square closed Lie ideal $𝔏$ of $𝔄$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $𝔄$, every generalized Jordan triple higher derivation of $𝔏$ into $𝔄$ is a generalized higher derivation of $𝔏$ into $𝔄$.

Nonlinear $*$-Lie higher derivations of standard operator algebras

Communications in Mathematics

Let $ℋ$ be an infinite-dimensional complex Hilbert space and $𝔄$ be a standard operator algebra on $ℋ$ which is closed under the adjoint operation. It is shown that every nonlinear $*$-Lie higher derivation $𝒟={\left\{{\delta }_{n}\right\}}_{n\in ℕ}$ of $𝔄$ is automatically an additive higher derivation on $𝔄$. Moreover, $𝒟={\left\{{\delta }_{n}\right\}}_{n\in ℕ}$ is an inner $*$-higher derivation.

On Jordan ideals and left $\left(\theta ,\theta \right)$-derivations in prime rings.

International Journal of Mathematics and Mathematical Sciences

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