Let R be an integral domain. All modules under discussion are torsion free unital left R-modules.
An R-module is completely decomposable if it is the direct sum of rank 1 submodules.
An R-module is separable if each finite set of its elements is contained in a completely decomposable direct summand.
An R-module is 1-separable if each element is contained in a completely decomposable direct summand.
For torsion free abelian groups, 1-separable implies separable.
Suppose that the countable direct product of R is separable. If an R-module is 1-separable, is it also separable?
Please check these links:
http://www.uio.no/studier/emner/matnat/math/MAT4250/h13/separabilitet.pdf
https://www.math.utah.edu/~singh/publications/singh_separable.pdf
I hope you will find them useful..
Thank you, Ms. Mostafa. I will try to confirm that the definitions of "separable" are consistent.
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