唯一分解的伪欧氏环的结构

THE STRUCTURE OF UNIQUE DECOMPOSITION PSEUDO-EUCLIDEAN RING

本文定义了唯一分解的伪欧氏环 R.设 K0 由 0和 R中所有可逆元素组成 ,xα≠ 0满足 δ(xα) =ωα,本文证明 K0 是体 ,R中任一元素可唯一表示为形如axn1 a1 … xnmαm,(a∈ K0 ,0≤ a1 <… <am)的项的有限和 .并证明在唯一分解的伪欧氏环 R中 ,δ(ab) =δ(b) +δ(a) ,( a,b∈ R) .最后 ,给出当 K0 为域时

In this paper, the concept of unique decomposition pseudo-Euclidean ring is introduced. Asume that R is a unique decomposition pseudo-Euclidean ring. Let K 0 is the set of 0 and all its units, then K 0 is a sfield. Let x α≠0, δ(x α)=ω α be arbitrary, it is proved that any element in R can be uniquely represented as the sum of finite items with the from as ax n 1 α 1...x n m α m,(0≤α 1<...<α m). It is proved also the δ(ab)=δ(b)+δ(a),(a,b∈R). In the end, we give the structure of R in the case of K 0 is a field.