关于有限环

On Finite Ring

本文研究具有ｎ（ｎ≥２）个左零因子，且Ｒ的元数大于ｎ的有限环，证明了如下定理：１．如果存在Ｒ的一个左零因子ａ，使Ｒａ＝Ｄｌ（此处Ｄｌ表示Ｒ的一切左零因子集），则Ｄｌ是Ｒ的幂零理想，并且Ｄｌ＝Ｊ（Ｒ）．当且仅当｜Ｒ｜＝ｎ２时，Ｄｌ的幂零指数为２．而当Ｄｌ的幂零指数为Ｋ时，存在素数ｐ以及自然数ｓ，使得ｎ＝ｐ（ｋ－１）ｓ且｜Ｒ｜＝ｎｍ，ｍ｜ｎ．２．对于任一非零左零因子ａ均有Ｒａ＝Ｄｌ的充要条件是｜Ｒ｜＝ｎ２．

Let R be a finite ring with n(n ≥2)left zero divisors and |R|>n. The following theorems are proved in this paper. 1)Let D l be the set of all left zero divisors of R .If there exists an element a of D l such that Ra=D l, then D l is a nilpotent ideal of R and D l=J(R). 2)In 1),| R|=n 2 is the necessary and sufficient condition such that the index of nilpotency of D l is 2. 3)In 1),for every non-zero left zero divisor a we have Ra=D l is the necessary and sufficient condition such that the index of nilpotency of D is 2. 4)In 1),if the index of nilpotency of D l is k,then there exists a nature number s and a prime number p such that n=p (k-1)s ,|R|=nm and m is a divisor of n . We pointed out some mistakes of literatures using above theorems.