Exact Sequences in (R,S)-Modules

Yuwaningsih, Dian Ariesta; Cahyo, Yassin Dwi; Subastian, Nanda

doi:10.20956/ersnw733

Open access Journal

Exact Sequences in (R,S)-Modules

Jurnal Matematika, Statistika dan Komputasi · Vol. 23 No. 1 (2026) · pp. 38–48

DOI: 10.20956/ersnw733
ISSN: 2614-8811, 1858-1382
Published: 2026-09-15
Publisher: Department of Mathematics, Hasanuddin University
Language: en
License: creativecommons.org/licenses/by/4.0

Abstract

The concept of modules has been generalized to the structure of (R,S)-bimodules, which has further been extended to (R,S)-modules. Various properties of classical module theory have been developed within the framework of (R,S)-modules. One of the important topics in module theory that can be extended to this setting is the concept of exact sequences. In this paper, we develop the notion of exact sequences in modules to exact sequences in (R,S)-modules. We begin by introducing the definitions of exact sequences and short exact sequences in (R,S)-modules and examine their basic properties. Several examples are provided to illustrate the concepts and to highlight the similarities and differences with classical module theory. In addition, we establish criteria for determining when a short exact sequence of (R,S)-modules splits and present related structural results using commutative diagrams. Furthermore, we present the necessary and sufficient conditions for an (R,S)-module to be Noetherian in a short exact sequence. These results contribute to a deeper understanding of the homological behavior of -modules and provide a foundation for further research in this area.

Keywords

Collaboration network of the authors

9 authors · 16 co-authorship links · 10 across different organisations

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Universitas Ahmad Dahlan
Universitas Gadjah Mada
Diponegoro University

Different organisations
Same organisation
Affiliation unknown
Thicker line = more joint articles · bigger circle = more articles

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Co-author	Organisation	Joint articles
Gusti Ayu Dwi Yanti	Universitas Gadjah Mada	2
Rima Aksen Cahdriyana	Universitas Ahmad Dahlan	2
Dea Apriani	Universitas Ahmad Dahlan	2
Fayi Salsabila Sumaryati	Universitas Ahmad Dahlan	1
Indah Emilia Wijayanti	Universitas Gadjah Mada	1
Budi Surodjo	Universitas Gadjah Mada	1

Affiliations come from OpenAlex; journal OAI-PMH metadata carries none. Authors without an OpenAlex ID are matched by name, which can merge or split people.

Metadata source: OAI-PMH (metadata jurnal) https://journal.unhas.ac.id/index.php/jmsk/oai, first harvested 5 Nov 2026, last seen 8 Nov 2026.

Original metadata (oai_dc)

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	<dc:title xml:lang="en">Exact Sequences in (R,S)-Modules </dc:title>
	<dc:creator xml:lang="en">Yuwaningsih, Dian Ariesta</dc:creator>
	<dc:creator xml:lang="en">Cahyo, Yassin Dwi</dc:creator>
	<dc:creator xml:lang="en">Subastian, Nanda</dc:creator>
	<dc:subject xml:lang="en">homomorphism</dc:subject>
	<dc:subject xml:lang="en">exact sequence</dc:subject>
	<dc:subject xml:lang="en">Noetherian</dc:subject>
	<dc:subject xml:lang="en">(RS)-modul</dc:subject>
	<dc:subject xml:lang="en">Abstract algebra</dc:subject>
	<dc:subject xml:lang="en">Algebra</dc:subject>
	<dc:subject xml:lang="en">Module Theory</dc:subject>
	<dc:description xml:lang="en">The concept of modules has been generalized to the structure of (R,S)-bimodules, which has further been extended to (R,S)-modules. Various properties of classical module theory have been developed within the framework of (R,S)-modules. One of the important topics in module theory that can be extended to this setting is the concept of exact sequences. In this paper, we develop the notion of exact sequences in modules to exact sequences in (R,S)-modules. We begin by introducing the definitions of exact sequences and short exact sequences in (R,S)-modules and examine their basic properties. Several examples are provided to illustrate the concepts and to highlight the similarities and differences with classical module theory. In addition, we establish criteria for determining when a short exact sequence of (R,S)-modules splits and present related structural results using commutative diagrams. Furthermore, we present the necessary and sufficient conditions for an (R,S)-module to be Noetherian in a short exact sequence. These results contribute to a deeper understanding of the homological behavior of -modules and provide a foundation for further research in this area.</dc:description>
	<dc:publisher xml:lang="en">Department of Mathematics, Hasanuddin University</dc:publisher>
	<dc:date>2026-09-15</dc:date>
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	<dc:identifier>https://journal.unhas.ac.id/index.php/jmsk/article/view/50504</dc:identifier>
	<dc:identifier>10.20956/ersnw733</dc:identifier>
	<dc:source xml:lang="en">Jurnal Matematika, Statistika dan Komputasi; Vol. 23 No. 1 (2026): September 2026; 38-48</dc:source>
	<dc:source>2614-8811</dc:source>
	<dc:source>1858-1382</dc:source>
	<dc:language>en</dc:language>
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	<dc:coverage xml:lang="en">mathematics</dc:coverage>
	<dc:rights xml:lang="en">Copyright (c) 2026 Jurnal Matematika, Statistika dan Komputasi</dc:rights>
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