主理想整环上的线性方程与一类矩阵方程

本文给出含幺主理想整环上线性方程组与一类矩阵方程可解的条件与通解。

Linear Equations and Some Kinds of Matrix Equation over a P.I.D

Let R be a principal ideal domain (P. I. D)with Unit 1.In this paper,we will give the solvableand the solution of linear equations in n unknowns and somekinds of matrix equation over R,the main results are foliowing:Theorem 1 If rank where the then:(1)AX=C is solVable if and only if is a factor of ,that is,there exist a such that where is defined by (2)If the solvable condition hold,and then isa peicular solution of Theorem 2 If AX=C is solvable,then the general solution of is of the form Where is a Particular solution ofis a base of Theorem 3 If diag where is the Kronecker product: then:(1)is solvable if and only if is a factor of PC=C1(2)If the solvable condition hold,thenis a particular solution of where is the isomorphism