Attaching topological spaces to a module (I): Sobriety and spatiality

On the De Morgan’s Laws for Modules

“…Remark 2.13. In the case A = Λ(M ) and B = Λ f i (M ), following [MBMCSMZC18] we write LgSpec(M ) = Spec B (A) and we call it the large spectrum of M and for Spec B (B) we just write Spec(M ). Note that when R = M , Spec(M ) is the usual prime spectrum.…”

“…If M is a self-progenerator in σ[M ], the frame Ψ(M ) is characterized as the fixed points of an operator named Ler : Properties of this operator are given in [MBMCSMZC18]. Also, in [MBMCSMZC20], more characterizations of this operator and the frame Ψ(M ) are obtained when M is a strongly harmonic or Gelfand module.…”

“…Following this way, in Section 5, we study properties of (N : L) for fully invariant submodules of a module M, we study the strong algebraic De Morgan's law, and finally, in Theorem 5.16 is given a characterization for a Noetherian prime ring to be an Asano prime ring. By last, it is worth to mention that this manuscript is a natural step in our investigation, that was initiated in [MBSMZC16] followed by [MBMCSMZC18] and [MBMCSMZC20].…”