Factorization Of Matrices Over Unique Factorization Domains

The research of multivariate polynomial rings and matrices over multivariate )olynomial rings is an important subject which arithmetic algebra and symbolic computation discuss. Finitely generated projective modules and many problems about algebra can come down to problems about matrices over multivariate polynomial ings. In 1976, Quillen and Suslin proved independently that finitely generated projective modules over k[xl,x1,…,xn](where k is a field) are free (Serre’s conjecture, 1955). It is equivalent to the proposition that any unimodular row (v1,v2,…,vm)(m≥3) over k[x1,x2…,xn]can be embedded into an invertible matrix over k[xl,x2,…xn]. Thus, esearch on finitely generated projective modules can be translated into that of natrices.In this paper, some properties of the resultant of two polynomials and some natrix factorization problems over elementary divisor integral domains or local rings re discussed. Specifically, in the third chapter some properties of the resultant of two )olynomials are discussed, I conclude that if f(X),g(X),u(X)∈R[X],o(f)≥1. +uf≠0,f is monic, then R(f,g+uf)=R(f,g). In the fourth chapter, I conclude some properties about exact division in K-Hermite rings and if R is an elementary ivisor integral domain, A∈Mm×n(R), rk(A)=r(r>0), then there exist P∈Mm×r(R),Q∈Mr×n(R) such that A=PQ, where the ranks of P and Q are r. In he fifth chapter, some matrix factorization problems over local rings, especially roblems about matrices being factored into the product of some elementary matrices ire discussed. I conclude that if R is a principal ideal domain, and A=(pq0rs0001)∈SL3(R[X]),then A∈E3(R[x]).