It is shown that: (1) every direct summand of a coatomic refinable module is coatomic refinable; (2) over a left max ring a module M is (strongly) coatomic refinable if and only if it is (strongly) refinable; (3) if a coatomic refinable module M is π-projective, then it is strongly coatomic refinable; (4) if coatomic direct summands lift modulo every coatomic submodule of M, then M is coatomic refinable.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In this article, we define the concept of (strongly) coatomic refinable modules as a proper generalization of (strongly) refinable modules.
📊
Key Findings & Data:In this article, we define the concept of (strongly) coatomic refinable modules as a proper generalization of (strongly) refinable modules.
💡
Conclusion & Impact:It is shown that: (1) every direct summand of a coatomic refinable module is coatomic refinable; (2) over a left max ring a module M is (strongly) coatomic refinable if and only if it is (strongly) refinable; (3) if a coatomic refinable module M is π-projective, then it is strongly coatomic refinable; (4) if coatomic direct summands lift modulo every coatomic submodule of M, then M is coatomic refinable.
In this article, we define the concept of (strongly) coatomic refinable modules as a proper generalization of (strongly) refinable modules. It is shown that: (1) every direct summand of a coatomic refinable module is coatomic refinable; (2) over a left max ring a module M is (strongly) coatomic refinable if and only if it is (strongly) refinable; (3) if a coatomic refinable module M is π-projective, then it is strongly coatomic refinable; (4) if coatomic direct summands lift modulo every coatomic submodule of M, then M is coatomic refinable.
Referans1 J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting modules supplements and projectivity in module theory, Frontiers in Mathematics, Birkhauser Basel, 2006. Referans2 A.Ç Ozcan, A. Harmancı, and P.F. Smith, Duo modules, Glasgow Mathematical Journal 48 (2006), 533-545. Referans3 S. H. Mohamed and B. J. Müller, Continuous and discrete modules, London Math. Soc. LNS 147 Cambridge University, Cambridge, 1990. Referans4 R. Wisbauer, Foundations of modules and rings theory, Gordon and Breach, 1991. Referans5 R. Wisbauer, Modules and algebras: bimodule structure and group actions on algebras, Pitman Monographs and Surveys in Pure and Applied Mathematics81, 1996. Referans6 H. Zoschinger, Komplementierte moduln uber Dedekindringen, J. Algebra 29 (1974), 42-56.