D3-MODULES VERSUS D4-MODULES - APPLICATIONS TO QUIVERS


D'este G., KESKİN TÜTÜNCÜ D., Tribak R.

GLASGOW MATHEMATICAL JOURNAL, vol.63, no.3, pp.697-723, 2021 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 63 Issue: 3
  • Publication Date: 2021
  • Doi Number: 10.1017/s0017089520000452
  • Journal Name: GLASGOW MATHEMATICAL JOURNAL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Computer & Applied Sciences, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.697-723
  • Keywords: 16D10, 16G20, MODULES
  • Hacettepe University Affiliated: Yes

Abstract

A module M is called a D4-module if, whenever A and B are submodules of M with M = A circle plus B and f : A -> B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X circle plus X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.