关于交换环上矩阵嵌入可逆矩阵的一些条件

Some Conditions for a Matrix to be Embeddable in an Invertible Matrix over a Commutative Ring

给出了交换环上一个矩阵可嵌入到可逆矩阵的一个必要条件和一个充分条件,进而证明了主理想整环上一个n阶矩阵可嵌入到一个n+1阶可逆矩阵的充要条件是这个矩阵的伴随矩阵的元素是互素的.部分结果推广了整数环上的结论.

A necessary condition and a sufficient condition for a matrix to be embeddable in an invertible matrix over a commutative ring are given.Furthermore,it is proved that a necessary and sufficient condition for a matrix of order nto be embeddable in an invertible matrix of order n+1 over a principal ideal domain is that the elements in the adjoint matrix of this matrix are relatively prime.Partial results obtained in this paper generalize the corresponding results for the ring of integers.