一般环之因子幂零理想

Factor-nilpotent Ideal of Rings

中文摘要:

      本文称环Ω的左(右)理想A为因子幂零的,如果对于任意元素r∈Ω,均有正整数m=m(r),使得A^mr={0}.称Ω的一个左理想L为关于元素b∈Ω的左因子,如果Lb≠{0}.定理4 设R是环Ω的因子幂零右理想,那么R+ΩR是Ω的一个因子幂零理想.定理7 设Ω具有局部左因子极小条件,那么Ω的任意诣零左理想必是因子幂零左理想.本文指出因子幂零性是介于幂零性与诣零性之间的一种性质,更接近幂零性。

英文摘要:

      Let n be a ring. A left (right) ideal A of ft is called factor-nilpotent if there is a positive integer m = m(r) with A^mr = {0} for every element r ∈Ω. A left ideal L of Ω is called a left factor for an element b ∈Ω, if Lb ≠ {0}.Ω is called a ring with locally minimum condition for left factors, if in fl every descending chain of left factors for the same element is finite. Here we show that1 Let R be a factor-nilpotent right ideal of Ω. Then R + ΩR is a factor-nilpotent ideal of Ω.2 Let Ω be a ring with locally minimum condition of left factors. Then every nil left ideal of Ω is a factor-nilpotent left ideal.