On Rings for Which Each Module With the Property (〖S_s〗^* ) Is an Extending Module

An S-module H is named s-cosingular if H=Z_s^* (H) ={h∈H├| Sh┤┤ is a semisimple and small ├ module}. An S-module H is said to satisfy the property (〖S_s〗^* ) 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 (〖S_s〗^* ) if 〖Z_s〗^* (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 (〖S_s〗^* )" and "each module with (〖S_s〗^* ) is injective".