一类特殊的有限循环环

证明了一类n阶(n=P_1P_2…p_m,p_i(i=1,2,…,m)互异为素数)环是有限循环环,并讨论了他们的结构及相关性质,最后给出了这类n阶环有零因子或有子域的充要条件.主要结果:P_1P_2…P_m阶环共有2^m个,它们是(p_(1m个,它们是(p_(1^k_1) p_(2k_1) p_(2^k_2)…p_(mk_2)…p_(m^k_m)Z)/(p_(1k_m)Z)/(p_(1^k_1+1)p_(2k_1+1)p_(2^k_2+1)…p_(mk_2+1)…p_(m^k_m+1)Z),其中k_i=0或1,1≤i≤m;阶是n=P_1P_2…p_m的环R可唯一分解为m个素数阶理想的直和,即R=〈α〉=(?);含pi(1≤i≤m)阶子域的P_1P_2…P_m阶环共有2k_m+1)Z),其中k_i=0或1,1≤i≤m;阶是n=P_1P_2…p_m的环R可唯一分解为m个素数阶理想的直和,即R=〈α〉=(?);含pi(1≤i≤m)阶子域的P_1P_2…P_m阶环共有2^(m-1)个,它们是p_(1(m-1)个,它们是p_(1^k_1) p_(2k_1) p_(2^k_2)…p_(mk_2)…p_(m^k_m)Z)/(p_(1k_m)Z)/(p_(1^k_1+1)p_(2k_1+1)p_(2^k_2+1)…p_(mk_2+1)…p_(m^k_m+1)Z),其.中k_i=0,k_j=0或1,1≤j≤m,j≠i.

A Special Sort of Finite Cyclic Rings

In this paper,we prove that a ring with order n(where n = p_1p_2…p_m,P_i(i = 1,2,…,m) are distinct primes) is a finite cycle ring.The structure of finite cycle rings and some related properties are discussed.Finally,some sufficient and necessary conditions for a finite cycle rings that have zero divisors or subfields are obtained.The main results:there are a total of 2~m rings with order p_1p_2…p_m,which are(p_1~(k_1)p_2~(k_2)…p_m~(k_m)Z)/(p_1~(k_1+1)p_2~(k_2+1)…p_m~(k_m+1)Z), where k_i= 0 or 1,1≤i≤m;the ring R with order n = p_1P_2…p_m can be uniquely decomposed into the direct sum of m prime order ideals,which is,R =〈a〉=(?)〈n/(p_i)a〉;there are a total of 2~(m-1) rings with orderp_1p_2…P_m which each has a subfield with order p_i(1≤i≤m), which are {p_1~(k_1)p_2~(k_2)…P_m~(k_m)Z)/(P_1~(k_1+1)p_2~(k_2+1)…P_m~(k_m+1)Z),where k_i= 0,k_j =0 or 1,1≤j≤m, j≠i.