| The generalized inverse theory is very important in matrix theory. Research on the com- mutative rings is almost perfectly, and the study on common filed, Divion Ring or PID are developping with different process.However, the generalized inverses over non-commutative ring haven't been discussed much. Bezout domain is a kind of non-commutative rings with great importancy. So it is useful to discuss the generalized inverse over Bezout domain. The partial ording of matrices is facous on the matrix theory,many mathematicians have been engaged in studying the partial ording of matrix such as kinds of partial ordering and its application.If every finitely generated left(right) ideal in a non-zero ring R which has a unit element 1 and no zero divisors is principal, then R is called Bezout domain.Integral Ring,Polynomial ring in an indeterminate over field,Division Ring,non-commutative P.I.D.,and Valuation Ring and so on are Bezout domain.The main purpose of this paper is to study the generalized inverse over Bezout domain, the necessary and sufficient conditions for the existence of the solutions of matrix equations AX = B, AX - YB = C,the partial ording over Bezout domain.In the third chapter, The definition of generalized inverse of matrix over Bezout domain is given. By using the definition of unitary matrix ,we get the necessary and sufficient conditions for exisitence of A+ , and {i ,…, j}-Inverses of its expressions if they exisit. The previous results are extended. Consequently some meaningful results are obtained.In the fourth chapter, we study the necessary and sufficient conditions for the existence of the solutions of matrix equations AX = B, AX - YB = C when A{ 1} exsists. Based on this, their equivalent characterizations are also obtained .In the fifth chapter, the concept of the partial ording over Bezout domain is given .The more precision characterization of the star, left-star, right star, sharp partial and the minus partial ordings are given repectively.
|
PDF Full Text Request