TARU PUBLICATIONS
Journal of Discrete Mathematical Sciences and Cryptography cover

Journal of Discrete Mathematical Sciences and Cryptography

(Since 1998)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

CiteScore 2025: 2.7 (Q1)

Powered by:Powered by

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Taru Online First Current IssueSpecial IssuesBrowse all Issues
Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

Unique ℱ-Closure Modules (𝒰ℱ𝒞 -module) and  ℱ-𝒢-extending modules

* ,

* Corresponding author · click or hover a name for details

pp. 2139–2146Vol. 29Issue 5May 2026DOI: 10.47974/JDMSC-2716 Crossmark XML
Received:
01 Jan 2026
Published Online:
22 May 2026
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2716
Pages:
2139–2146

Abstract

In this research, we presented new types of modules, which are ℱ -extending modules,  ℱ-Goldie extending (ℱ-𝒢-extending module) and a unique ℱ-closure module (𝒰ℱ𝒞 -module), Which are generalizations for each of extending modules, Goldie extending module and a unique closure module.   In this work, we introduce a new type of modules that is ψF-G-extending modules and we define a module M is called ψF-G-Goldie-extending iff for all submodule 𝒱 of M that containing F, where 𝓗 ≤F⊕ M, F ≤ 𝓗  satisfies  and V ψF 𝓗. We provide many characteristics related to this kind. In a module M, we define the relationships on the group of submodules of M: i. Vψℱ B  iff 𝒦 ≤ M   satisfies 𝒱 ≤ℱE 𝒦 and  B ≤ℱE 𝒦.  ii. Vφℱ B iff  𝒱 ∩ B ≤FE 𝒱 and 𝒱 ∩ B ≤FE B. A module ℳ is called ℱ -Goldie extending (ℱ-𝒢-extending module) if for every 𝒱 ≤ ℳ that contains ℱ, where a ℱ-direct summand 𝓗 of ℳ satisfies ℱ ≤ 𝓗 and 𝒱 φℱ 𝓗  and we introduce some Characteristics and Features for this kind module.

Keywords

F-essential submodulesF-closedF-G extending

Subject Classifications

03C3003C5203C5003C90

References

[1] E. Akalan, G. F. Birkenmeier, and A. Tercan, “Goldie extending modules,” Communications in Algebra, vol. 37, no. 2, pp. 663–683 (2009), doi: 10.1080/00927870802254843.
[2] N. V. Dung, D. V. Huynh, P. F. Smith, and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics Series, no. 313. Harlow, UK/New York, USA: Longman Scientific & Technical / Chapman & Hall/CRC (1994).
[3] A. Shihadeh, “An approach to Goldie extending modules on the class of cyclic submodules,” Communications on Applied Nonlinear Analysis, vol. 31, no. 3S, pp. 9–17 (2024), doi: 10.52783/cana.v31.727.
[4] P. Fleury, “Hollow modules and local endomorphism rings,” Pacific Journal of Mathematics, vol. 53, no. 2, pp. 379–385 (1974).
[5] F. Sh. Fandi, “On essential T-Goldie* lifting modules,” AIP Conference Proceedings, vol. 2834, pp. 080126-1–080126-5 (2023), doi: 10.1063/5.0161707.
[6] R. Beyranvand and F. Moradi, “Small submodules with respect to an arbitrary submodule,” Journal of Algebra and Related Topics, vol. 3, no. 2, pp. 43–51 (2015).
[7] F. Sh. Fandi and S. M. Yaseen, “Essential T-hollow-lifting module,” Journal of Physics: Conference Series, vol. 1530, no. 1, Art. no. 012070 (2020), doi: 10.1088/1742-6596/1530/1/012070.

Views: 34Downloads: 9Citations: 0