SSEP Modules and Trivial Extensions

Eren Doğan; Mustafa Alkan

doi:10.54559/amesia.1736730

SSEP Modules and Trivial Extensions

Abstract

In this paper, we study the behavior of modules possessing the Summand Sum Essential Property (SSEP) in the context of trivial extensions of rings. We provide a detailed examination of how the SSEP and the classical Summand Sum Property (SSP) are preserved or characterized when extending a ring by an (R, R)-bimodule. Necessary and sufficient conditions are established under which the trivial extension inherits these properties from the base ring. Our results generalize existing findings on summand properties and contribute to a deeper understanding of module decompositions and essentiality in extended algebraic structures. Several illustrative examples are provided to demonstrate the applicability of the theoretical developments.

Keywords

References

J. L. Garcia, Properties of direct summands of modules, Communications in Algebra 17 (1) (1989) 73-92.
M. Alkan, A. Harmancı, On summand sum and summand intersection property of modules, Turkish Journal of Mathematics 26 (2) (2002) 131-147.
A. N. Abyzov, T. H. N. Nhan, T. C. Quynh, Modules close to SSP and SIP modules, Lobachevskii Journal of Mathematics 38 (1) (2017) 16-23.
I. Amin, Y. Ibrahim, M. F. Yousif, D3-modules, Communications in Algebra 42 (2) (2014) 578-592.
F. W. Anderson, K. R. Fuller, Rings and categories of modules, Springer, New York, 1974.
D. M. Arnold, J. Hausen, A characterization of modules with the summand intersection property, Communications in Algebra 18 (2) (1990) 519-528.
G. F. Birkenmeier, F. Karabacak, A. Tercan, When is the SIP (SSP) property inherited by free modules, Acta Mathematica Hungarica 112 (1-2) (2006) 103-106.
J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting modules: Supplements and projectivity in module theory, Birkhäuser, Basel, 2006.

N. V. Dung, D. V. Huynh, P. F. Smith, R. Wisbauer, Pitman research notes in mathematics, Vol. 313, Extending modules, Longman Scientific and Technical, 1994.
L. Fuchs, Infinite abelian groups, Academic Press, New York-London, 1970.
K. R. Goodearl, Ring theory: Nonsingular rings and modules, CRC Press, 1976.
G. Güngöroğlu, A. Harmancı, On some classes of modules, Czechoslovak Mathematical Journal 50 (125) (2000) 839-846.
J. Hausen, Modules with the summand intersection property, Communications in Algebra 17 (1) (1989) 135-148.
I. Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954.
F. Karabacak, A. Tercan, Matrix rings with summand intersection property, Czechoslovak Mathematical Journal 53 (128) (2003) 621-626.
F. Kasch, Modules and rings, Academic Press, 1982.
Ö. Taşdemir, F. Karabacak, Generalized SIP-modules, Hacettepe Journal of Mathematics and Statics 48 (4) (2019) 1137-1145.
Ö. Taşdemir, F. Karabacak, Generalized SSP-modules, Communications in Algebra 48 (3) (2020) 1068-1078.
T. Y. Lam, Graduate texts in mathematics: Lectures on modules and rings, Springer, 2012.
S. H. Mohammed, B. J. Müller, Continuous and discrete modules, Vol. 147, London mathematical society lecture note series, Cambridge University Press, Cambridge, 1990.
R. N. Roberts, Krull dimension for Artinian modules over quasi-local commutative rings, Quarterly Journal of Mathematics 26 (1) (1975) 269-273.
Y. Talebi, A. R. M. Hamzekolaee, On a generalization of lifting modules via SSP-modules, Bulletin of the Iranian Mathematical Society 48 (2) (2022) 343-349.
G. V. Wilson, Modules with the summand intersection property, Communications in Algebra 14 (1) (1996) 21-38.
E. Doğan, M. Alkan, Modules and matrix rings with SIEP and SSEP properties, Montes Taurus Journal of Pure Applied Mathematics (in press).

Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Eren Doğan ^*
0000-0003-0027-2930
Türkiye

Mustafa Alkan
0000-0002-4452-4442
Türkiye

Publication Date

December 31, 2025

Submission Date

July 7, 2025

Acceptance Date

September 5, 2025

Published in Issue

Year 2025 Volume: 6 Number: 2

DOI

https://doi.org/10.54559/amesia.1736730

IZ

https://izlik.org/JA77TK86RC

Cite

RIS / Bibtex

APA

Doğan, E., & Alkan, M. (2025). SSEP Modules and Trivial Extensions. Amesia, 6(2), 82-89. https://doi.org/10.54559/amesia.1736730

AMA

1.Doğan E, Alkan M. SSEP Modules and Trivial Extensions. Amesia. 2025;6(2):82-89. doi:10.54559/amesia.1736730

Chicago

Doğan, Eren, and Mustafa Alkan. 2025. “SSEP Modules and Trivial Extensions”. Amesia 6 (2): 82-89. https://doi.org/10.54559/amesia.1736730.

EndNote

Doğan E, Alkan M (December 1, 2025) SSEP Modules and Trivial Extensions. Amesia 6 2 82–89.

IEEE

[1]E. Doğan and M. Alkan, “SSEP Modules and Trivial Extensions”, Amesia, vol. 6, no. 2, pp. 82–89, Dec. 2025, doi: 10.54559/amesia.1736730.

ISNAD

Doğan, Eren - Alkan, Mustafa. “SSEP Modules and Trivial Extensions”. Amesia 6/2 (December 1, 2025): 82-89. https://doi.org/10.54559/amesia.1736730.

JAMA

1.Doğan E, Alkan M. SSEP Modules and Trivial Extensions. Amesia. 2025;6:82–89.

MLA

Doğan, Eren, and Mustafa Alkan. “SSEP Modules and Trivial Extensions”. Amesia, vol. 6, no. 2, Dec. 2025, pp. 82-89, doi:10.54559/amesia.1736730.

Vancouver

1.Eren Doğan, Mustafa Alkan. SSEP Modules and Trivial Extensions. Amesia. 2025 Dec. 1;6(2):82-9. doi:10.54559/amesia.1736730

EBSCO	DOAJ
Scilit	SOBIAD