利用剩余类环的性质探索行列式中的一个概率问题

Utilizing the Properties of Residue Class Ring to Solve Several Probability Problems in Determinant

设一个n阶行列式,其中每一个元素都以等概率取{kr+i∶k∈Z},(i=0,1,2,…,r-1)中的整数,其中r>1为任意的正整数。该行列式的值除以r的余数会以一定的概率规律取集合{0,1,2,…,r-1}中的元素。本文利用近世代数中的剩余类环的相关理论就这个问题进行了一些有益的探索,并获得了一些结论。

Let M be a n-determinant whose entries take the numbers from the set-{kr＋i∶k∈Z},（i=0,1,2,…,r-1） with the equal probability,where r is larger than 1,and randomly takes a positive number.Making the value of that determinant to divide r,the remainder will take the elements from the set-{0,1,2,…,r-1}according to a certain law.This paper,on the basis of the related theories of Residue Class Ring from Abstract Algebra,makes an effort to initiate the instrumental research pointing to this problem and,consequently,achieves several conclusions.