有限局部环Z/q^kZ上矩阵广义逆的几个计数结果

Some Anzahl Theorems of Generalized Inverse Matrices Over Z/q^kZ

设 R =Z/ qk Z是模整数 qk的有限局部环 ,其中 q是素数 ,k>1 .对 R上给定的 n阶矩阵 A,设 W1={X∈ Mn( R) |PAXP- 1=Q- 1XAQ, 1 P,Q∈ GLn( R) },W2 ={X∈ Mn( R) |AX =XA},W3={X∈ Mn( R) |AXA =A},W4 ={X∈ Mn( R) |XAX =X}.若 Wi≠Φ( i=1 ,2 ,3 ,4) ,用 n( Wi)表示 Wi中所有元素的个数 ,主要计算出 n( Wi) ( i =1 ,2 ,3 ,4)

Let R=Z/qkZ is a finite local ring of module integer qk, where q is a prime and k>1 If matrix A is the matrix be given over R, let W 1={X∈M n(R)-PAXP -1=Q -1XAQ, 1P, Q∈GL n(R)}, W 2={X∈M n(R)-AX=XA}, W 3={X∈M n(R)-AXA=A}, W 4={X∈M n(R)-XAX=X}. In this Paper, n(W i) are computed, where W i≠Φ and n(W i) denote the numbers of elements in W i, i=1,2,3,4.