Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$, $C$ the extended centroid of $R$ and $a\in R$. Suppose that $\delta$ is a nonzero $\sigma$-derivation of $R$ such that $a{[\delta \left(x^n\right),x^n]}_k=0$ for all $x\in R$, where $\sigma$ is an automorphism of $R$, $n$ and $k$ are fixed positive integers. Then $a=0$.

Let $R$ and $S$ be commutative rings with unity, $f:R\to S$ a ring homomorphism and $J$ an ideal of $S$. Then the subring $R{\bowtie }^{f}J:=\{(a,f\left(a\right)+j)\mid a\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we determine when $R{\bowtie }^{f}J$ is a (generalized) filter ring.

Let $ℛ$ be a semiprime ring with unity $e$ and $\phi$, $\varphi$ be automorphisms of $ℛ$. In this paper it is shown that if $ℛ$ satisfies $$2𝒟\left(x^n\right)=𝒟\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒟\left(x^{n-1}\right)$$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ is an ($\phi$, $\varphi$)-derivation. Moreover, this result makes it possible to prove that if $ℛ$ admits an additive mappings $𝒟,𝒢:ℛ\to ℛ$ satisfying the relations $$\begin{array}{c}2𝒟\left(x^n\right)=𝒟\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒢\left(x\right)+𝒢\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒢\left(x^{n-1}\right),\\ 2𝒢\left(x^n\right)=𝒢\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒟\left(x^{n-1}\right),\end{array}$$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ and $𝒢$ are ($\phi$, $\varphi$)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras. ...

For $k\in \mathbb{N}\cup \left\{\infty \right\}$ and $U$ open in $ℝ$, let ${\mathrm{C}}^k\left(U\right)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f\in {\mathrm{C}}^k\left(U\right)$ there is $g\in {\mathrm{C}}^{\infty }\left(ℝ\right)$ with $U\subseteq \mathrm{coz}g$ and $h\in {\mathrm{C}}^k\left(ℝ\right)$ with ${fg|}_U{=h|}_U$. The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$. The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring ${\mathrm{C}}^k\left(ℝ\right)$ are deduced from this, in particular that ${\mathrm{Q}}_{\mathrm{cl}}\left({\mathrm{C}}^k\left(ℝ\right)\right)=\mathrm{Q}\left({\mathrm{C}}^k\left(ℝ\right)\right)$.

Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\ge 1$ a fixed positive integer. Let $\alpha$ be an automorphism of the ring $R$. An additive map $D:R\to R$ is called an $\alpha$-derivation (or a skew derivation) on $R$ if $D\left(xy\right)=D\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. An additive mapping $F:R\to R$ is called a generalized $\alpha$-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F\left(xy\right)=F\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. We prove...

Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha$, and $p(x_1,...,x_n)$ is a non-central polynomial over $C$ such that $$\left[F\right(x),\alpha (y\left)\right]=G\left(\right[x,y\left]\right)$$ for all $x,y\in \{p(r_1,...,r_n):r_1,...,r_n\in R\}$. Then there exists $\lambda \in C$ such that $F\left(x\right)=G\left(x\right)=\lambda \alpha \left(x\right)$ for all $x\in R$.

Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_2$ acting on $H$ by inverting elements, where $C_2$ is the cyclic group of order two. Let $\Omega \left(𝒟\right)$ be the Burnside ring of $𝒟$, $\Delta \left(𝒟\right)$ be the augmentation ideal of $\Omega \left(𝒟\right)$. Denote by ${\Delta }^n\left(𝒟\right)$ and $Q_n\left(𝒟\right)$ the $n$th power of $\Delta \left(𝒟\right)$ and the $n$th consecutive quotient group ${\Delta }^n\left(𝒟\right)/{\Delta }^{n+1}\left(𝒟\right)$, respectively. This paper provides an explicit $\mathbb{Z}$-basis for ${\Delta }^n\left(𝒟\right)$ and determines the isomorphism class of $Q_n\left(𝒟\right)$ for each positive integer $n$.

A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J\left(R\right)$ is nil and $R/J\left(R\right)$ is $*$-tripotent. Furthermore, we explore...

We characterize clean elements of $ℛ\left(L\right)$ and show that $\alpha \in ℛ\left(L\right)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that ${𝔠}_L\left(\mathrm{coz}(\alpha -\mathbf{1})\right)\subseteq U\subseteq {𝔬}_L\left(\mathrm{coz}\left(\alpha \right)\right)$. Also, we prove that $ℛ\left(L\right)$ is clean if and only if $ℛ\left(L\right)$ has a clean prime ideal. Then, according to the results about $ℛ\left(L\right),$ we immediately get results about ${𝒞}_c\left(L\right).$

A metacyclic group $H$ can be presented as $\langle \alpha ,\beta :{\alpha }^n=1$, ${\beta }^m={\alpha }^t$, $\beta \alpha {\beta }^{-1}={\alpha }^r\rangle$ for some $n$, $m$, $t$, $r$. Each endomorphism $\sigma$ of $H$ is determined by $\sigma \left(\alpha \right)={\alpha }^{x_1}{\beta }^{y_1}$, $\sigma \left(\beta \right)={\alpha }^{x_2}{\beta }^{y_2}$ for some integers $x_1$, $x_2$, $y_1$, $y_2$. We give sufficient and necessary conditions on $x_1$, $x_2$, $y_1$, $y_2$ for $\sigma$ to be an automorphism.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J\left(M_2\left(R\right)\right)$, or $I_2-A\in J\left(M_2\left(R\right)\right)$, or $A$ is similar to $\left(\begin{array}{cc}0& \lambda \\ 1& \mu \end{array}\right)$, where $\lambda \in J\left(R\right)$, $\mu \in 1+J\left(R\right)$, and the equation $x^2-x\mu -\lambda =0$ has a root in $J\left(R\right)$ and a root in $1+J\left(R\right)$. We further prove that $f\left(x\right)\in R\left[\right[x\left]\right]$ is strongly $J$-clean if $f\left(0\right)\in R$ be optimally $J$-clean.