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A ring R is semiperfect if and only if, for every (cyclic)R-module M, every maximal submodule has (ample) supplements in M. A non-local commutative domain R is h-local if and only if, for every (cyclic) torsion R-module M, every maximal submodule of M has (ample) supplements in M.
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Ali̇zade et al. (2001) studied this question.
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