关于整数环的无限n素元组归纳扩环的多样性  

ON THE ABUNDANCE OF THE INDUCTIVE EXTENSIONS OF THE RING OF INTEGERS WITH INFINITELY MANY n-PRIME TUPLES

作　　者：别荣芳[1] 

机构地区：[1]北京师范大学信息科学学院,北京100875

出　　处：《北京师范大学学报（自然科学版）》2005年第4期339-342,共4页Journal of Beijing Normal University(Natural Science)

基　　金：国家自然科学基金重点资助项目(19931020);国家自然科学(青年)基金资助项目(60273015;10001006)

摘　　要：用模型论方法证明了:对于整数环I及每个正整数n及每一可能类型的n素元组而言,存在不可数无限多个其1阶性质互不全同的含有无限多该类型的n素元组的归纳扩环(文中简称该类型的nT-环),并且存在无限多个正整数对,对每个这样的正整数对(a,b),存在I的nT-扩环R,它适合加a的归纳法,而不适合加b的归纳法.还证明了存在该类型的nT-环R,R的每个元素是3平方和和4立方和.It is shown by model theoretic methods that the following results are valid for any positive integer n and any possible type of n-prime tuples： （1） There exist uncountably infinitely many inductive extensions of the ring I of integers with infinitely many such n-prime tuples（called nT-rings in the following） and these rings are not equivalent to each other in first order logic. （2） There exist infinitely many couples of positive integers such that for each couple（a, b）, there exists an nT-ring R which satisfies the inductive principle with step a and does not satisfy the inductive principle with step b. （3） There exist nT-rings R such that every element of R is a sum of 3 squares and a sum of 4 cubes.

关 键 词：归纳环 n素元组 模型论 

分 类 号：O141.4[理学—数学]