On a question concerning D4-modules

Authors

  • Soumitra Das Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore-641407, India

DOI:

https://doi.org/10.21638/spbu01.2021.308

Abstract

An R-module M is called a D4-module if ‘whenever M1 and M2 are direct summands of M with M1 + M2 = M and M1 ∼= M2, then M1 ∩ M2 is a direct summand of M’. Let M = ⊕i∈IMi be a direct sum of submodules Mi with Hom(Mi,Mj ) = 0 for distinct i, j ∈ I. We show that M is a D4-module if and only if for each i ∈ I the module Mi is a D4-module. This settles an open question concerning direct sums of D4-modules. Our approach is independent of the solution obtained by D’Este, Keskin T¨ut¨unc¨u and Tribak recently.

Keywords:

SIP-modules, D4-modules

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References

Литература/References

1. Kaplansky I. Infinite Abelian Groups. Ann Arbor, Univ. of Michigan Press (1969).

2. Garcia J.L. Properties of direct summands of modules. Comm. Algebra 17, 73–92 (1989). https://doi.org/10.1080/00927878908823714

3. Hausen J. Modules with the summand intersection property. Comm. Algebra 17, 135–148 (1989). https://doi.org/10.1080/00927878908823718

4. Wilson G.V. Modules with the direct summand intersection property. Comm. Algebra 14, 21–38 (1986).

5. Alkan M., Harmanci A. On summand sum and summand intersection property of modules. Turkish J. Math. 26, 131–147 (2002).

6. Mohamed S., Muller B.J. Continuous and Discrete Modules. In: London Math. Soc. Lecture Note Ser., vol. 147. Cambridge, Cambridge Univ. Press (1990).

7. Ding N., Ibrahim Y., Yousif M., Zhou Y. D4-modules. J. Algebra Appl. 16 (5), 1750166 (2017). https://doi.org/10.1142/S0219498817501663

8. Clark J., Lomp C., Vanaja N., Wisbauer R. Lifting modules: supplements and projectivity in module theory. Basel, Boston, Berlin, Birkh¨auser, Verlag (2006).

9. Baba Y., Oshiro K. Classical Artinian Rings and Related Topics. World Scientific Publishing (2009).

10. Nguyen X.H., Yousif M., Zhou Y. Rings whose cyclics are D3-modules. J. Algebra Appl. 16 (8), 1750184 (2017). https://doi.org/10.1142/S0219498817501845

11. Ibrahim Y. Email communication (2020).

12. D’Este G., Keskin Tutuncu D., Tribak R. D3-modules versus D4-modules - Applications to quivers. Glasgow Math. Journal, 1–27 (2020). https://doi.org/10.1017/S0017089520000452

13. Lee G., Rizvi S.T., Roman C.S. Rickart modules. Comm. Algebra 38 (11) 4005–4027 (2010). https://doi.org/10.1080/00927872.2010.507232

14. Lee G., Rizvi S.T., Roman C.S. Dual Rickart modules. Comm. Algebra 39 (11), 4036–4058 (2011). https://doi.org/10.1080/00927872.2010.515639

15. Abyzov A.N., Tuganbaev A.A. Modules in which sums or intersections of two direct summands are direct summands. J. Math. Sciences 211 (3), 297–303 (2015). https://doi.org/10.1007/s10958-015-2605-0

16. Ding N., Ibrahim Y., Yousif M., Zhou Y. C4-modules. Comm. Algebra 45 (4), 1727–1740 (2017). https://doi.org/10.1080/00927872.2016.1222412

17. Altun-Ozarslan M., Ibrahim Y., Ozcan A.G., Yousif M. C4- and D4-Modules via perspective direct summands. Comm. Algebra 46 (10), 4480–4497 (2018). https://doi.org/10.1080/00927872.2018.1448838

18. Anderson F.W., Fuller K.R. Rings and Categories of Modules. In: Graduate Texts in Math., vol. 13. New York, Springer-Verlag (1974).

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Published

2021-09-26

How to Cite

Das, S. (2021). On a question concerning D4-modules. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(3), 467–474. https://doi.org/10.21638/spbu01.2021.308

Issue

Vol. 8 No. 3 (2021)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.