A ring $R$ is called right P-injective if every homomorphism from a principal right ideal of $R$ to $R_R$ can be extended to a homomorphism from $R_R$ to $R_R$. Let $R$ be a ring and $G$ a group. Based on a result of Nicholson and Yousif, we prove that the group ring $\mathrm{RG}$ is right P-injective if and only if (a) $R$ is right P-injective; (b) $G$ is locally finite; and (c) for any finite subgroup $H$ of $G$ and any principal right ideal $I$ of $\mathrm{RH}$, if $f\in {\mathrm{Hom}}_R(I_R,R_R)$, then there exists $g\in {\mathrm{Hom}}_R({\mathrm{RH}}_R,R_R)$ such that ${g|}_I=f$. Similarly, we also obtain equivalent...

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 ${𝒞}_c\left(L\right).$

In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$-semicommutative. We prove that a ring $R$ is $P$-semicommutative if and only if $R\left[x\right]$ is $P$-semicommutative if and only if $R[x,x^{-1}]$ is $P$-semicommutative. Also, if $R\left[\right[x\left]\right]$ is $P$-semicommutative, then $R$ is $P$-semicommutative. The converse holds provided that $P\left(R\right)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$, $R$ is $P$-semicommutative...

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 $a=0$.

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 $ℛL$ be the ring of real-valued continuous functions on a frame $L$. The aim of this paper is to study the relation between minimality of ideals $I$ of $ℛL$ and the set of all zero sets in $L$ determined by elements of $I$. To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$, it is proved that the $f$-ring $ℛL$ is isomorphic to the $f$-ring $C\left(\Sigma L\right)$ of all real continuous functions on the topological space $\Sigma L$. Finally, a one-one correspondence...

The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let $K$ be a unital commutative ring, not necessarily a field. Given a unital $K$-algebra $S$, where $K$ is contained in the center of $S$, $n\in \mathbb{N}$, the goal of this paper is to study the question: when can a homomorphism $\phi :{\mathrm{M}}_n\left(K\right)\to {\mathrm{M}}_n\left(S\right)$ be extended to an inner automorphism of ${\mathrm{M}}_n\left(S\right)$? As an application of main results presented in the paper, it is proved that if $S$ is...

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 $f:A\to B$ 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 $(f,g)$.

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. ...