## Currently displaying 1 – 12 of 12

Showing per page

Order by Relevance | Title | Year of publication

### 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 , , , ,  and ) 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  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

Page 1