摘要
设Ｒ是一结合环，Ｎ表示Ｒ的所有幂零元形成的集合．本文证明了：定理如果环Ｒ满足条件：对于任意ｘ＿１…，ｘ＿ｋ∈Ｒ，存在依赖于ｘ＿１，…，ｘ＿ｋ的字ω＿１＝ω＿１（ｘ＿２，…，ｘ＿ｋ）和ω＿２＝ω＿２（ｘ＿１，…，ｘ＿（ｋ－１））满足｜ω＿１｜ｘ＿ｋ＞１，｜ω｜＿ｘ＿１＞１，和｜ω＿１｜≠｜ω＿２｜，使得ｘ＿１ω＿１（ｘ＿２，…，ｘ＿ｋ）＝ω＿２（ｘ＿１，…，ｘ＿（ｋ－１））ｘ＿ｋ，这里ｋ＞１是一固定整数，则（ｉ）Ｎ形成Ｒ的理想；（ｉｉ）Ｒ＝ＮＲ＿１，这里Ｒ＿１同构于一些域的次直和．特别地，如果Ｎ是交换的，则Ｒ也是交换的。
Let R be a ring, and N the set of all nilpotent elements of R. We prove thefollowingTheorem if a rinq R satisfies the following condition: for all x_1…,x_k in R, there existwords ω_2 = ω_1(x_2,…,x_k ) and ω_2 = ω_2(x_1…,x_(k-1) depending on x_1 …,x_k such that|ω_1|x_k> 1,|ω_2|>1, and|ω_1|≠|ω_2|. Suppose thatx_1 ω_1 (x_2…,x_k ) = ω_2 ( x_1…,x_(k-1))x_kwhere k >1 is a fixed integer. Then(1) N jorms an ideal of R,(2) R = N + R_1, where R_1 is isomorphic to a subdirect sum of fields. In particular, ifN is co mmutative, then R is commutative.
关键词
环
结合环
幂零元
Rings, subdirect sum, words.
