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 $(n-1,n)$-...

in this paper, we introduce a new generalization of weakly prime ideals called $i$-prime. suppose $r$ is a commutative ring with identity and $i$ a fixed ideal of $r$. a proper ideal $p$ of $r$ is $i$-prime if for $a, b in r$ with $ab in p-ip$ implies either $a in p$ or $b in p$. we give some characterizations of $i$-prime ideals and study some of its properties.  moreover, we give conditions  ...

‎Let $G$ be a group with identity $e.$ Let $R$ be a $G$-graded‎ ‎commutative ring and $M$ a graded $R$-module‎. ‎In this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎For example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎Also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $M$ are studied‎.‎

The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...

In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...

Let $A$ be a Banach algebra and $E$ be a Banach $A$-bimodule then $S = A oplus E$, the $l^1$-direct sum of $A$ and $E$ becomes a module extension Banach algebra when equipped with the algebras product $(a,x).(a^prime,x^prime)= (aa^prime, a.x^prime+ x.a^prime)$. In this paper, we investigate $triangle$-amenability for these Banach algebras and we show that for discrete inverse semigroup $S$ with...

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prim...

In this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. We prove that if $N$ is a weakly prime subtractive ternary subsemimodule of a ternary $R$-semimodule $M$, then either $N$ is a prime ternary subsemimodule or $(N : M)(N : M)N = 0$. If $N$ is a $Q$-...