Abstract: In this paper, left quasi- dual bimodules can be charac terized as those bimodules for which any submodule K of M_R and any left idea L of _SS are a direct summand of rM l_S (K ) and l_S r_M ( L ) respectly. For a left quasi- dual bimodule _SM_R, some results are proved: ( 1) _SM is a Kasch module, ( 2) r_M l_S ( Soc( M_R ) ) = Soc(M_R  ) and l_S r_M ( Soc( _SS ) ) = Soc( _SS ) , ( 3) l_S( Soc( M_R  ) )  J ( S) and r_M ( Soc( _SS ) ) Rad( M_R  ) , ( 4) if M_R  is a CS- module, the Soc
( M_R  )  _eM_R , ( 5) if M_R  is nonsingular, then M is semisimple, ( 6) if M_R  is project ive in [ M] and M_R  is semisimple, the M is a nonsingular module. It deduced that if R is a left dualring or a left quasi- dual ring, then R is a semisimple if R is nonsingular.

Key words: quasi- dual bimodules, M- singular modules, Kasch module, dual rings