Infinite System Of Equations On The Division Ring

Division ring is an important algebraic system, due to thenon-commutativity of multiplication, the investigation of the division ring ismore difficult than that of the field, and the cognition of the division ring isnot much hitherto. In (8), Xie investigated the matrices and the systems ofequations on division ring, and presented the existence theorem and thestructure theorem of the solutions .In (7), Li obtained the Core of the matriceson division ring ----the nilpotent decomposition theorem. In (3), Wangstudied the inverse matrix and the diagonalization of infinity matrices onfield and obtained some significative results. In (1), (2) and (3), Cheninvestigated the infinity matrices on division ring, and obtained the conditionof the existence of one-sided inverse matrix of infinity matrix and thedecomposition theorem of infinity matrix, and investigated thediagonalization of infinity matrices on division ring simultaneously. Based onthese results, we investigate a kind of systems of equations formed by finiteequations of finite variables, and obtained the existence theorem and thestructure theorem of the solutions.Theorem 1 SupposeAX=O (*)is an infinite system of equations, the right maximal linearly independentsubset of A form a matrix B, if B has the left inverse matrix of finite numberof rows, and the number of the column vectors of the matrix which formedby omitting the set of the column vectors of B from the set of the columnvectors of A is n, then(1) the only solution of (*) is zero while n=0;(2) the fundamental system of solutions of (*) is formed by n vectorswhile n is a finite number;(3) the fundamental system of solutions of (*) is formed by countablevectors while n is infinite.Theorem 1 The general solution of the system of non-homogeneousequations is the summation of a special solution of the system and the generalsolution of the corresponding system of homogeneous equations.We shall note that, despite existent of the left inverse matrix of B, therow number of the left inverse matrix of B may be infinite. In this case, weshall discuss correctness of the foregoing theorems and the structure of thesolutions otherwise.