拟对偶双边模与对偶环

左拟对偶双边模 _SM_R 可以被刻划成M_R 的任意子模K 和_SS 的任意左理想L 分别是_rM l_S (K ) 和 l_S r_M( L ) 的一个直和项.对一个左拟对偶双边模_SM_R, 有以下结论: ( 1) _SM 为Kasch模; ( 2) r_Ml_S ( Soc( M_R ) ) = Soc(M_R ) , l_S r_M ( Soc( _S^S) ) = Soc( _SS) ;( 3) l_S ( Soc(M_R ) ) J ( S) , r_M ( Soc( _SS) ) Rad(M_R ) ; ( 4) 若 M_R 为 CS- 模,则 Soc( M^R ) _eM_R ; ( 5) 若 M_R 是非M - 奇异的,则M 是半单的; ( 6) 若 M_R 在[ M] 中投射且 M_R 半单,则 M 是非M - 奇异模.并且还得出, 若 R 是左对偶环或左拟对偶环,则R 是半单环当且仅当R 非奇异.

Quasi-dual Bimodules and Dual Rings

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.