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D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS

Part of: Representation theory of rings and algebras Modules, bimodules and ideals

Published online by Cambridge University Press:  06 October 2020

GABRIELLA D′ESTE ,
DERYA KESKİN TÜTÜNCÜ  and
RACHID TRIBAK
GABRIELLA D′ESTE
Affiliation:
Department of Mathematics, University of Milano, Milano, Italy e-mail: gabriella.deste@unimi.it
DERYA KESKİN TÜTÜNCÜ
Affiliation:
Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey e-mail: keskin@hacettepe.edu.tr
RACHID TRIBAK
Affiliation:
Centre Régional des Métiers de l’Education et de la Formation (CRMEF-TTH)-Tanger, Avenue My Abdelaziz, Souani, B.P. 3117, Tangier, Morocco e-mail: tribak12@yahoo.com
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Abstract

A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ 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 ⊕ 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.

MSC classification

Primary: 16D10: General module theory
Secondary: 16G20: Representations of quivers and partially ordered sets
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 3 , September 2021 , pp. 697 - 723
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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