The purpose of this paper is to study topological properties both the set all $k$-prime ideals and congruences for any commutative semiring with zero identity. We first prove that spectrum, i.e. equipped Zariski topology a spectral space, then homeomorphic spectrum respect their topologies.

Let $M$ and $N$ be two finitely generated graded modules over a standard graded Noetherian ring $R=bigoplus_{ngeq 0} R_n$. In this paper we show that if $R_{0}$ is semi-local of dimension $leq 2$ then, the set $hbox{Ass}_{R_{0}}Big(H^{i}_{R_{+}}(M,N)_{n}Big)$ is asymptotically stable for $nrightarrow -infty$ in some special cases. Also, we study the torsion-freeness of graded generalized local ...

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...

For a module M over commutative ring R with identity, let RSpec(M) denote the collection of all submodules L Msuch that ?(L:M) is prime ideal and equal to (rad L:M). In this article, we topologies topology which enjoys analogs many properties Zariski on spectrum Spec(M) (as subspace topology). We investigate topological space from point view spectral spaces by establishing interrelations betwee...

The prime k-spectrum Speck(R) of a k-semiring R will be introduced. It will be proven that it is a topological space, and some properties of this space will be investigated. Connections between the topological properties of Speck(R) and possible algebraic properties of the k-semiring R will be established.

Let F be a field, let D be a subring of F , and let X be the Zariski-Riemann space of valuation rings containing D and having quotient field F . We consider the Zariski, inverse and patch topologies on X when viewed as a projective limit of projective integral schemes having function field contained in F , and we characterize the locally ringed subspaces of X that are affine schemes.