Hostname: page-component-6766d58669-bp2c4 Total loading time: 0 Render date: 2026-05-24T16:55:59.075Z Has data issue: false hasContentIssue false
  • English
  • Français

D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS

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

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
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Altun-Özarslan, M., Ibrahim, Y., Özcan, A. Ç. and Yousif, M., C4- and D4-modules via perspective direct summands, Comm. Alg. 46 (2018), 4480–4497.CrossRefGoogle Scholar
Auslander, M., Reiten, I. and Smalϕ, S. O., Representation theory of artin algebras (Cambridge University Press, UK, 1997).Google Scholar
Azumaya, G., Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem, Nagoya Math. J. 1 (1950), 117–124.CrossRefGoogle Scholar
Castro Pérez, J., Medina Bárcenas, M., Ríos Montes, J. and Zaldívar, A., On semiprime Goldie modules, Comm. Alg. 44 (2016), 4749–4768.CrossRefGoogle Scholar
Castro Pérez, J. and Ríos Montes, J., Prime submodules and local Gabriel correspondence in σ[M], Comm. Alg. 40 (2012), 213–232.CrossRefGoogle Scholar
Clark, J., Lomp, C., Vanaja, N. and Wisbauer, R., Lifting modules. Supplements and projectivity in module theory, Frontiers in Mathematics (Birkhäuser, Basel-Boston-Berlin, 2006).Google Scholar
Ding, N., Ibrahim, Y., Yousif, M. F. and Zhou, Y., D4-modules, J. Alg. Appl. 16(9) (2017), 1750166.CrossRefGoogle Scholar
Fuchs, L. and Rangaswamy, K. M., Quasi-projective abelian groups, Bull. Soc. Math. France 98 (1970), 5–8.CrossRefGoogle Scholar
Goodearl, K. R., Ring theory: Nonsingular rings and modules (Marcel Dekker, Inc. New York and Basel, 1976)Google Scholar
Goodearl, K. R. and Warfield, R. B., Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16 (Cambridge University Press, Cambridge, 1989).Google Scholar
Haghany, A. and Vedadi, M. R., Study of semi-projective retractable modules, Alg. Coll. 14 (2007), 489–496.CrossRefGoogle Scholar
Idelhadj, A. and Tribak, R., On some properties of ⊕-supplemented modules, IJMMS 69 (2003), 4373–4387.Google Scholar
Kaplansky, I., Infinite Abelian groups (The University of Michigan Press, Ann Arbor, 1954).Google Scholar
Kaplansky, I., Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340.CrossRefGoogle Scholar
Keskin Tütüncü, D., B. Kalebog̃az and P. F. Smith, Direct sums of semi-projective modules, Colloquium Mathematicum 127 (2012), 67–81.CrossRefGoogle Scholar
Keskin Tütüncü, D., Orhan Ertaş, N. and Tribak, R., On weak Rickart modules, J. Alg. Appl. 16(9) (2017), 1750165.CrossRefGoogle Scholar
Krylov, A. and Tuganbaev, A. A., Modules over discrete valuation domains (Walter de Gruyer, New York, 2008).CrossRefGoogle Scholar
Lee, G., Rizvi, S. T. and Roman, C., Rickart Modules, Comm. Alg. 38 (2010), 4005–4027.CrossRefGoogle Scholar
Mohamed, S. H. and Müller, B. J., Continuous and discrete modules (Cambridge University Press, Cambridge, UK, 1990).CrossRefGoogle Scholar
Nicholson, W. K., Semiregular modules and rings, Canad. J. Math. XXVIII(5) (1976), 1105–1120.CrossRefGoogle Scholar
Rangaswamy, K. M. and Vanaja, N., Quasi-projectives in Abelian and module categories, Pacific J. Math. 43 (1972), 221–238.CrossRefGoogle Scholar
Ringel, C. M., Tame algebras and integral quadratic forms (Springer LMN 1099, 1984).CrossRefGoogle Scholar
Ringel, C. M., Infinite length modules. Some examples as introduction, in Infinite length modules (Bielefeld, 1998), Trends in Mathematic, vol. 173 (Birkhäuser, Basel, 2000).CrossRefGoogle Scholar
Schiffler, R., Quiver representations (Springer International Publishing, Switzerland, 2014).Google Scholar
Skowronski, A. and Yamagata, K., Frobenius algebras I: Basic representation theory, EMS Textbooks in Mathematics (European Mathematical Society, Switzerland, 2012).Google Scholar
Warfield, R. B. Jr., A Krull-Schmidt theorem for infinite sums of modules, Proc. Amer. Math. 22 (1969), 460–465.CrossRefGoogle Scholar
Yousif, M., Amin, I. and Ibrahim, Y., D3-modules, Comm. Alg. 42 (2014), 578–592.CrossRefGoogle Scholar