Exact Sequences in (R,S)-Modules
DOI:
https://doi.org/10.20956/ersnw733Keywords:
homomorphism, exact sequence, Noetherian, (RS)-modulAbstract
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.
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