On the second spectrum and the second classical Zariski topology of a module

Ceken S., ALKAN M.

JOURNAL OF ALGEBRA AND ITS APPLICATIONS, vol.14, no.10, 2015 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 14 Issue: 10
  • Publication Date: 2015
  • Doi Number: 10.1142/s0219498815501509
  • Journal Name: JOURNAL OF ALGEBRA AND ITS APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: Second submodule, second spectrum, Zariski topology, second classical Zariski topology, NONCOMMUTATIVE RINGS, PRIME SPECTRUM, DUAL NOTION, SUBMODULES
  • Akdeniz University Affiliated: Yes

Abstract

Let R be an associative ring with identity and Spec(s)(M) denote the set of all second submodules of a right R-module M. In this paper, we investigate some interrelations between algebraic properties of a module M and topological properties of the second classical Zariski topology on Spec(s)(M). We prove that a right R-module M has only a finite number of maximal second submodules if and only if Spec(s)(M) is a finite union of irreducible closed subsets. We obtain some interrelations between compactness of the second classical Zariski topology of a module M and finiteness of the set of minimal submodules of M. We give a connection between connectedness of Spe(c)s(M) and decomposition of M for a right R-module M. We give several characterizations of a noetherian module M over a ring R such that every right primitive factor of R is artinian for which Spec(s)(M) is connected.