环Z[X]不是中华环(英文)

The Ring Z[X] is not Chinese

中文摘要:

      设R是具有单位元1的交换环;A是R中的理想而a,b则是R中的任意元.定义a≡b(A)若Ra+A=Rb+A.称环R是中华环若a≡b(A+B),则存在c∈R使c≡a(A)及c≡b(B).环是中华环的充要条件是由K.Aubert与A.Beck二人于1980年找出的.显然,整数环Z必是中华环.Aubert与Beck二人亦证明了Z[x,y]不是中华环.但他们二人无法证明Z[X]是否中华环.本文用不同的手法处理,证明了Z[X]不可能是中华环.同时,我们进一步证明,对任意代数数a,环Z[a]均是中华环.因此,Aubert与Beck在1980年所提出的问题,在本文中得到圆满的解答.

英文摘要:

      A commutative ring R is said to be Chinese if, given a, b 6 R and ideals A, B of R such that a ≡ b(A + B), there exists c ∈ R such that c ≡ a(A) and c ≡ b(B). Chinese rings were investigated by K.E.Aubert and I.Beck in 1982. However, in their paper, they said that they were unable to settle the case whether the ring Z[X] is Chinese or not. In this paper, we provide a short proof to show that the ring Z[X] is not Chinese. The techinque we used here is different from Aubert and Beck. Moreover, we show that for any algebraic numbers α₁,…,α_n, the ring Z|α₁,…,α_n|is Chinese for n≥1.