归纳环及归纳域

提出了归纳环的概念(即:含1且适合1阶Peano归纳公理的环),指出其意义并提出下列关于确定归纳环代数结构的问题:是否每一归纳环都与由整数环的一族剩余类环及特征数0的代数闭域所作的一个超积初等等价?首先给出了有关归纳环及归纳域的一些基本事实.然后着重在代数范围内进行讨论,得到下列的结果:每个有限次实代数数域都不是归纳域;代数整数环的每个子环都不是归纳环.还证明了很多有限次代数数域不与上述的超积初等等价.

ON INDUCTIVE RINGS AND INDUCTIVE FIELDS

The author defines the concept of an inductive ring(that is, a ring con- taining 1 and satisfying the first-order Peano induction axioms). The author points out its significance, and raises the following problem for the deter- mination of the structure of inductive rings: Is it true that every inductive ring must be elementarily equivalent to an ultraproduct of a family of residue class rings of the rational integers and algebraically closed fields of characteristic zero? Firstly, some basic facts about inductive rings and ind- uctive fields are given. Then he considers the case of algebraic numbers and obtains some results like the following: Every real algebraic number field of finite degree is non-inductive; Every subring of the ring of algebraic in- tegers is non-inductive. Also, it is proved that many algebraic number fields of finite degree are not elementarily equivalent to the above-mentioned ultraproducts.