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...

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...

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.

In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let $J^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of $J^{\#}$. A ring $R$ is said to be in case every element in $R$ is very $J^{\#}$-clean. We prove that every very $J^{\#}$-clean ring is strongly $\pi$-rad clean and has stable range one. It is shown that for a...

For a finite group $G$ and a fixed Sylow $p$-subgroup $P$ of $G$, Ballester-Bolinches and Guo proved in 2000 that $G$ is $p$-nilpotent if every element of $P\cap G^{\text{'}}$ with order $p$ lies in the center of $N_G\left(P\right)$ and when $p=2$, either every element of $P\cap G^{\text{'}}$ with order $4$ lies in the center of $N_G\left(P\right)$ or $P$ is quaternion-free and $N_G\left(P\right)$ is $2$-nilpotent. Asaad introduced weakly pronormal subgroup of $G$ in 2014 and proved that $G$ is $p$-nilpotent if every element of $P$ with order $p$ is weakly pronormal in $G$ and when $p=2$, every element of $P$ with...

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 $n$.

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.

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 $A\left(X\right)$ denote a subalgebra of $C\left(X\right)$ which is closed under local bounded inversion, briefly, an $LBI$-subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A\left(X\right)$, we generalize the notion of $z_A^{\beta }$-ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C\left(X\right)$...

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 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 $a=0$.