Quasi-dual Bimodules and Dual Rings

In this paper, left quasi-dual bimodules can be characterized as those bimodules for which any submodule K of M R and any left idea L of SS are a direct summand of r Ml S (K) and l Sr M (L) respectly. For a left quasi-dual bimodule S M R, some results are proved:(1) SM is a Kasch module,(2)r Ml SWTB(Soc(M R))=Soc(M R) and l Sr M(Soc(X] SS))=Soc( SS),(3)l S(Soc(M R))J(S) and r M(Soc( SSWTB))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 projective in σ SS))=Soc( SS),(3)l S(Soc(M R))J(S) and r M(Soc( SSWTB))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 projective in σ and M R is semisimple, the M is a nonsingular module. It deduced that if R is a left dual ring or a left quasi-dual ring, then R is a semisimple if R is nonsingular.