二次数环的不具有Goldbach性质的扩环

EXTENSIONS OF QUADRATIC RINGS OF ALGEBRAIC INTEGERS WITHOUT GOLDBACH PROPERTY

在[1]中,我们通过考虑2次代数整数环的某些剩余类环并引用模型论中的紧致性定理,证明了:对每一2次代数整数环 J,都存在 J 的扩环,它适合 Goldbach 性质.(即:每一非0非单位的元α,其2倍都可表示为两个素元的和.)在[2]中。

By using number-theoretic and model-theoretic arguments similar to those used by the first author in [1] and [2],we proved in this paper the following theorem: Theorem 1 For any ring J of quadratic integers,there exists an exten- sion ring R of J which has the following properties:(a)R is a commutative ring with unity 1.(b) R has infinitely many prime elements.(c) R has in- finitely many composite elements.(d_1)R does not satisfy Goldbach property. (i.e.,there exists a non-zero and non-unit element a of R such that 2 a is not the sum ot two prime elements of R.) For comparison,we cite here the following theorem which follows easily from the results of [1]: Theorem 1′ For any ring J or quadratic integers,there exists an exten- 24 sion ring R of J which has the following properties:(a)R is a commutative ring with unity 1.(b)R has infinitely many prime elements.(c)R has infi- nitely many composite elements.(d_1′)R satisfies Goldbach property. Comparison of the above two theorems illustrates the independence of Goldbach property in various obvious (though rather weak) senses. Finally,it is pointed out that the following two much stronger theorems can be proved by using the lemmas of this paper and those of[1]: Theorem 2 For any ring J of quadratic integers,ther exists an exten- sion ring R of J which has the above properties (a),(b),(c)and the follo- wing: (d_2)There exists a non-zero,non-unit element aR such that for every po- sitive integer n,na is not the sum of n prime elements of R. Theorem 2′ For any ring J of quadratic integers,there exists an exten- sion ring R of J which has the above properties (a),(b),(c) and the follo- wing: (d_2′)For every non-zero,non-unit element a(?)R and every positive integer n>1, na is the sum of n prime elements of R. Note that each of (d_2),(d_2′) is much stronger than the negation of the other.