再论实闭环

FURTHER NOTES ON REAL CLOSED RINGS

继作者[4]与[7]中的工作对实闭环作进一步的探讨。主要的结果为定理1与3。另外,利用所获得的结果对文[3]中最后所提出的问题作了正面的解答,见定理5。

This paper continues an investigation by the auther in [3] and [4] on real closed rings.The main results are the following:Theorem 1.Let R be a real closed ring,and A be a proper subring of R.Then A is real closed if and only if it is integrally closed in R.Tehroem 4.If R is a real closed ring,then the total quotient ring T(R) of R is regular in the sense of von Neumann.Besides,as an application of Theorem 1,we give an affirmative answer to a problem raised at the end of [1],which is the following .Theorem 5.\ Let R be an integral domain having a non-trivial real valuation v with core (o) and valuation pair (A,M).R is then a real closed domain if and only if the following conditions are satisfied:1) (A,M) is algebraically maximal;2) A/M is a real closed domain;3) the value group of v is divisible.