Geodesic
D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS
Glasgow mathematical journal, Tome 63 (2021) no. 3, pp. 697-723

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DOI

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. 
D′ESTE, GABRIELLA; TÜTÜNCÜ, DERYA KESKİN; TRIBAK, RACHID. D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS. Glasgow mathematical journal, Tome 63 (2021) no. 3, pp. 697-723. doi: 10.1017/S0017089520000452
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     author = {D'ESTE, GABRIELLA and T\"UT\"UNC\"U, DERYA KESK\.IN and TRIBAK, RACHID},
     title = {D3-MODULES {VERSUS} {D4-MODULES} {\textendash} {APPLICATIONS} {TO} {QUIVERS}},
     journal = {Glasgow mathematical journal},
     pages = {697--723},
     year = {2021},
     volume = {63},
     number = {3},
     doi = {10.1017/S0017089520000452},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089520000452/}
}
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