Let $G$ be a monoid with identity $e$. In this paper, first we introduce the notions of $G$-graded hyperrings, graded hyperideals and graded hyperfields in the sense of Krasner hyperring $R$. Also, we define the notion of a greded $R$-hypermodules and some examples are presented. Then we investigate graded maximal, graded prime and graded primary hyperideals of a graded hyperring $R$. Finally, ...

In this paper we consider the problem of finding upperbounds on the minimum norm of representatives in residue classes in quotient O/I, where I is an integral ideal in the maximal order O of a number field K. In particular, we answer affirmatively a question of Konyagin and Shparlinski, stating that an upperbound o(N(I)) holds for most ideals I, denoting N(I) the norm of I. More precise stateme...

We consider maximal abelian subalgebras of O n which are invariant to the standard circle action. It turns out that these are all contained in the zero grade of O n. Then we consider shift invariant maximal abelian subalgebras of the zero grade, which are also invariant to a " second shift " map, and show that these are just infinite tensor products of diagonal matrices in the standard UHF pict...

the theorem 12 in [a note on $p$-nilpotence and solvability of finite groups, j. algebra 321(2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indice. in this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index aresolvable.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every st...

the notions of quasi-prime submodules and developed  zariski topology was introduced by the present authors in cite{ah10}. in this paper we use these notions to define a scheme. for an $r$-module $m$, let $x:={qin qspec(m) mid (q:_r m)inspec(r)}$. it is proved that $(x, mathcal{o}_x)$ is a locally ringed space. we study the morphism of locally ringed spaces induced by $r$-homomorphism $mrightar...

Let R be a commutative ring with nonzero identity and M an R-module. In this paper, first we give some relations between S-prime S-maximal submodules that are generalizations of prime maximal submodules, respectively. Then construct topology on the set all , which is generalization spectrum M. We investigate when SpecS(M) T0 T1-space. also study continuous maps irreducibility SpecS(M). Moreover...

In this paper, persents the definitions of strongly prime ideal, strongly prime N-subgroup, Pseudo-valuation near ring and Pseudo-valuation N-group. Some of their properties have also been proven by theorems. Then it is shown that, if N be near ring with quotient near-field K and P be a strongly prime ideal of near ring N, then is a strongly prime ideal of ‎‎, for any multiplication subset S of...

Let B/Q be an indefinite quaternion algebra of discriminant D (we allow the case of D = 1, i.e., B ∼= M2(Q)) and ON a level N Eichler order in B. By taking Γ(ON ) = Γ(B,ON ) to be the elements of O× of positive reduced norm, we get an arithmetic Fuchsian group and hence a Riemann surface ON\H. So as not to prejudice matters, let us temporarily denote the Shimura curve associated to any quaterni...