Modules and the second classical Zariski topology


Çeken S., ALKAN M.

Matematiche, vol.73, no.1, pp.161-177, 2018 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 73 Issue: 1
  • Publication Date: 2018
  • Doi Number: 10.4418/2018.73.1.12
  • Journal Name: Matematiche
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
  • Page Numbers: pp.161-177
  • Keywords: Second submodule, second spectrum, prime submodule, second classical Zariski topology, NONCOMMUTATIVE RINGS, PRIME SPECTRUM, DUAL NOTION, SUBMODULES
  • Akdeniz University Affiliated: Yes

Abstract

© 2018 Universita di Catania.Let R be an associative ring with identity and Specs(M) denote the set of all second submodules of a right R-module M. In this paper, we present a number of new results for the second classical Zariski topology on Specs(M) for a right R-module M. We obtain a characterization of semisimple modules by using the second spectrum of a module. We prove that if R is a ring such that every right primitive factor of R is right artinian, then every non-zero submodule of a second right R-module M is second if and only if M is a fully prime module. We give some equivalent conditions for Specs(M) to be a Hausdorff space or T1-space when the right R-module M has certain algebraic properties. We obtain characterizations of commutative Quasi-Frobenius and artinian rings by using topological properties of the second classical Zariski topology. We give a full characterization of the irreducible components of Specs(M) for a non-zero injective right module M over a ring R such that every prime factor of R is right or left Goldie.