【摘要】 给出了模型论在代数上的两个应用,得到了下列定理:定理A:如环R的任何有限生成子环均是局部环,则R是局部环.定理B:存在自然数的真扩张R使其具有下列特征:(1)虽然R有无限多零因子,但R中有无限多零因子,但R中的首1多项式的根的个数可以得到很好的控制.(2)R不仅将自然数的素数特征保留下来,而且还可在其上定义指数函数。更多还原

【Abstract】 This paper gives two applications of model theory in algebra:(A) If any finitely-generated subring of R is local,then R itself is local.(B) There are real extensions R of N which have the following properties:(1) Although R has infinite divisors of zero,but the number of roots of the polynomials with unit head coefficient is well controlled.(2) R not only preserves the properties of natural prime number,but also admits to define the exponent function in it.更多还原