An S-module H is named s-cosingular if H=Zₛ^* (H) =h∈H├| Sh┤┤ is a semisimple and small ├ module. An S-module H is said to satisfy the property (〖Sₛ〗^*) if H has a direct summand T for any submodule K of H such that T≤K and K/T is an s-cosingular module. It has been proved that for an extending module H exhibiting (〖Sₛ〗^*) if 〖Zₛ〗^* (H) is projective, then each submodule of H is an extending module and H is a locally artinian serial module. Semisimple and left ss-Harada rings have also been characterized by the conditions "each extending module exhibits (〖Sₛ〗^*) " and "each module with (〖Sₛ〗^*) is injective".
Kır et al. (Mon,) studied this question.
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