We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the ex...We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the existence and the expression of the general solution tothe system are derived. As applications, necessary and sufficient conditions are given for thesystem of matrix equations A_1X = C_1 and A_3X = C_3 to have a bisymmetric solution, the system ofmatrix equations A_1X = C_1 and A_3XB_3 = C_3 to have a perselfconjugate solution over R with aninvolution and char R≠2, respectively. The representations of such solutions are also presented.Moreover, some auxiliary results on other systems over R are obtained. The previous known results onsome systems of matrix equations are special cases of the new results.展开更多
In this paper, some new relations between GP-V'-rings and regular rings under certain special conditions have been found. It is proved that R is left self-injective regular with Soc(RR) ≠ 0 if and only if R is a ...In this paper, some new relations between GP-V'-rings and regular rings under certain special conditions have been found. It is proved that R is left self-injective regular with Soc(RR) ≠ 0 if and only if R is a left GP-V'-ring containing an injective maximal left ideal and Soc(RR)(?)Soc(RR). Moreover, for an MELT ring R, it is shown that R is regular if and only if R is a left GP-injective left GP-V'-ring.展开更多
基金This research is supported by the Natural Science Foundation of China(No.0471085the Natural Science Foundation of Shanghai)the Development Foundation of Shanghai Educational Committee the Special Funds for Major Specialities of Shanghai Education Co
文摘We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the existence and the expression of the general solution tothe system are derived. As applications, necessary and sufficient conditions are given for thesystem of matrix equations A_1X = C_1 and A_3X = C_3 to have a bisymmetric solution, the system ofmatrix equations A_1X = C_1 and A_3XB_3 = C_3 to have a perselfconjugate solution over R with aninvolution and char R≠2, respectively. The representations of such solutions are also presented.Moreover, some auxiliary results on other systems over R are obtained. The previous known results onsome systems of matrix equations are special cases of the new results.
基金This work was partially support by the NNSF of China (No. 10171011) the NSF of JiangsuProvince in China (No. BK 2001001) the Younger Foundation (2003xqn04) of Anhui Normal University.
文摘In this paper, some new relations between GP-V'-rings and regular rings under certain special conditions have been found. It is proved that R is left self-injective regular with Soc(RR) ≠ 0 if and only if R is a left GP-V'-ring containing an injective maximal left ideal and Soc(RR)(?)Soc(RR). Moreover, for an MELT ring R, it is shown that R is regular if and only if R is a left GP-injective left GP-V'-ring.
