Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$. In this manuscript, we investigate the action of skew derivation $(\delta ,\varphi )$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$. Moreover, we provide an example for semiprime case.

Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F\left(u\right){[F\left(u\right),u]}^n=0$ for all $u\in L$, where $n\ge 1$ is a fixed integer. Then one of the following holds: (1) ...

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

Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ${{(}^F)}^{\left[p^m\right]}{=}^{\left[p^m\right]}$, where ${}^F$ is the Frobenius closure of . This paper is concerned with the question whether the set $Q{(}^{\left[p^m\right]}):m\in \mathbb{N}₀$ is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map $R/{\sum }_{j=1}^{n}R{c}_j\to R/{\sum }_{j=1}^{n}Rc{²}_j$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in ass(c{₁}^j,...,c{ₙ}^j)$, c₁/1,..., cₙ/1 is a $R_{}$-filter...

Given a group X we study the algebraic structure of the compact right-topological semigroup λ(X) consisting of all maximal linked systems on X. This semigroup contains the semigroup β(X) of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup λ(X) in the semigroup ${\left(X\right)}^{\left(X\right)}$ of all self-maps of the power-set (X) and show that the image of λ(X) in ${\left(X\right)}^{\left(X\right)}$ coincides with the semigroup $En{d}_{\lambda }\left(\left(X\right)\right)$ of all functions f: (X) → (X) that are equivariant, monotone and symmetric...

Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $K$ and $K^{\text{'}}$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}\left(K\right)=g^{-1}\left(K^{\text{'}}\right)$. We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring $A{\bowtie }^{f,g}(K,K^{\text{'}})$ of $A$ with $(B,C)$ along $(K,K^{\text{'}})$ with respect to $(f,g)$.

At first we prove some results on a general polynomial derivation using few results of linear derivation. Then we study the ring of constants of a linear derivation for some rings. We know that any linear derivation is a nonsimple derivation. In the last section we find the smallest integer $w>1$ such that the polynomial ring in $n$ variables is $w$-differentially simple, all $w$ derivations are nonsimple and the $w$ derivations set contains a linear derivation.