一类整环中素元及唯一分解的研究

On Prime Elements and Unique Decomposition in a Class of Integral Rings

首先,把整环I中的整数表示为两个元的乘积,通过对整数取共轭,得到“如果其中一个元的虚部是0,那么另一个元的虚部也是0”。其次,利用近世代数中关于素元的理论知识,得到k±√2k+1i、小于w的素数和√w i都是整环I中的素元。最后,给出了I为唯一分解环的充分条件是w=2。

The integer is expressed as the product of two elements in the integral ring.By taking the conjugate of the integer,it is derivable that if the imaginary part of one element is 0,the imaginary part of the other element is also 0.Using the theoretical knowledge of prime elements and unique decomposition in modern algebra,we conclude that k±√2k+1i,prime numbers(<w)and √w i are prime elements in I.Finally,the sufficient condition for being a unique decomposition ring is given.