On Divided Modules


Tekir Ü., Ulucak G., Koç S.

IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY TRANSACTION A-SCIENCE, vol.44, no.A1, pp.265-272, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 44 Issue: A1
  • Publication Date: 2020
  • Doi Number: 10.1007/s40995-020-00827-1
  • Title of Journal : IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY TRANSACTION A-SCIENCE
  • Page Numbers: pp.265-272
  • Keywords: Divided ring, Divided module, Trivial extension

Abstract

Recall that a commutative ring R is said to be a divided ring if its each prime ideal P is comparable with each principal ideal (a), where a is an element of R. In this paper, we extend the notion of divided rings to modules in two different ways: let R be a commutative ring with identity and M a unital R-module. Then M is said to be a divided (weakly divided) module if its each prime submodule N of M is comparable with each cyclic submodule Rm (rM) of M, where m is an element of M (r is an element of R). In addition to give many characterizations of divided modules, some topological properties of (quasi-) Zariski topology of divided modules are investigated. Also, we study the divided property of trivial extension R proportional to M.