Modules Whose St-Closed Submodules are Fully Invariant
DOI:
https://doi.org/10.21123/bsj.2024.11434Keywords:
Closed submodule, Duo module, Fully Invariant submodule, St-closed submodule, Stc-duo module.Abstract
The duo module plays an important role in the module theory. Many researchers generalized this concept such as Ozcan AC, Hadi IMA and Ahmed MA. It is known that in a duo module, every submodule is fully invariant. This paper used the class of St-closed submodules to work out a module with the feature that all St-closed submodules are fully invariant. Such a module is called an Stc-duo module. This class of modules contains the duo module properly as well as the CL-duo module which was introduced by Ahmed MA. The behaviour of this new kind of module was considered and studied in detail, for instance, the hereditary property of the St-duo module was investigated, as the result; under certain conditions, every St-closed submodule of an St-duo module is also St-duo. Another characterization of the Stc-duo module was given. Additionally, the relationships of St-duo among some types of modules were investigated and discussed, for example; In the class of semi-extending modules, every weak duo module is an Stc-duo module. Also, the authors gave a case in which St-duo, duo, CL-duo and weak duo are equivalent. Furthermore, the St-duo module was used to make the concepts semi-extending and FI-extending equivalent.
Received 21/04/2024
Revised 12/07/2024
Accepted 14/07/2024
Published Online First 20/10/2024
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