let $r$ be a commutative ring with identity‎. ‎a proper ideal $p$ of $r$ is a $(n-1,n)$-$phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin r$‎, ‎$a_1cdots a_nin pbackslash p^m$ ($a_1cdots a_nin pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin p$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎in this paper several results concerning $(n-1,n)$-$phi_m$-prime and $(n-1,n)$-...

Starting from the difficulty of creating playful representation of domain-specific abstract concepts, this study discusses the design of Prime Slaughter, a computer game aimed at facilitating individual sense-making of abstract mathematical concepts. Specifically the game proposes a transpositionmathematical concepts. Specifically the game proposes a transposition of primality and factorization...

Throughout this paper, every groups are finite. The prime graph of a group $G$ is denoted by $Gamma(G)$. Also $G$ is called recognizable by prime graph if for every finite group $H$ with $Gamma(H) = Gamma(G)$, we conclude that $Gcong H$. Until now, it is proved that if $k$ is an odd number and $p$ is an odd prime number, then $PGL(2,p^k)$ is recognizable by prime graph. So if $k$ is even, the r...

We consider the problem of electing a leader on a ring of nameless processors by de-terministic and self-stabilizing protocols. A processor can read the state of its neighbors (and its own state) to determine if it is enabled. A central demon (scheduler) picks an enabled processor to make an atomic move. In an atomic move, the processor changes to a new state which is a function of its old stat...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

In the $B^+\to D^+ D^-K^+$ decay, LHCb has reported observation of open-charm exotic states $X_{0,1}^0\equiv X_{0,1}(2900)^0$ with four different quark flavors~$(ud\bar s \bar c)$, where subscripts (0,1) denote spins. To confirm discovery, we propose $\Lambda_b\to \Sigma_c^{0(++)} X_{0,1}^{\prime\,0(--)}$ in final state interaction, $X_{0,1}^{\prime\,0(--)}$ $s u\bar d c$ ($ds\bar u c$) are new...

We consider the problem of electing a leader on a ring of nameless processors by de terministic and self stabilizing protocols A processor can read the state of its neighbors and its own state to determine if it is enabled A central demon scheduler picks an enabled processor to make an atomic move In an atomic move the processor changes to a new state which is a function of its old state and th...