Quasi-prime Submodules and Developed Zariski Topology

Abstract

Let R be a commutative ring with nonzero identity and M be an R-module. Quasi-prime submodules of M and the developed Zariski topology on qSpec(M ) are introduced. We also, investigate the relationship between the algebraic properties of M and the topological properties of qSpec(M ). Modules whose developed Zariski topology is respectively T 0 , irreducible or Noetherian are studied, and several characterizations of such modules are given.

On Quasi Comultiplication Modules Over Pullback Rings

“…Proposition 3.3. Let R be the pullback ring as described in (1) and S be a non-zero separated R-module withS = 0. Then (0 :…”

“…Theorem 3.7. Let R be the pullback ring as described in (1). Then the indecomposable separated quasi comultiplication modules over R are: we must have S i = E(R i /P i ) or R i /P n i (n ≥ 1).…”

“…Let R be a pullback ring as described in(1) and M , S be two Rmodules. Let ϕ : S −→ M be an epimorphism.…”

Prime Submodules and a Sheaf on the Prime Spectra of Modules

“…Zariski topology on the spectrum of prime ideals of a ring is one of the main tools in algebraic geometry. In the literature, there are many different generalizations of the Zariski topology of rings to modules via prime submodules (see [3,8,19,27]). We recall that for any element r of a ring R, the set D r = Spec R \V Rr is open in Spec R and the family D r r ∈ R forms a base for the Zariski topology on Spec R .…”

“…Topologies are considered by Duraivel, McCasland, Moore, Smith, and Lu in [11,19,27]. In the literature, there are many papers devoted to the Zariski topology on the spectrum of modules [2,3,5,8,22,28]. It is well-known 3064 HASSANZADEH-LELEKAAMI AND ROSHAN-SHEKALGOURABI that Zariski topology on the spectrum of prime ideals of a ring is one of the main tools in algebraic geometry.…”