一类环的结构

The Structure of a Class of Rings

英文关键词:Rings  subdirect sum  words.

中文摘要:

      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（ｘ₂，…，ｘ_k）和ω₂＝ω₂（ｘ_１，…，ｘ_k-１）满足｜ω_１｜ｘ_k＞１，｜

英文摘要:

      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 x₁…,x_k in R, there existwords ω₂ = ω₁(x₂,…,x_k ) and ω₂ = ω₂(x₁…,x_k-1 depending on x₁ …,x_k such that|ω₁|x_k> 1,|ω₂|>1, and|ω₁|≠|ω₂|. Suppose that x₁ ω₁ (x₂…,x_k) = ω₂( x₁…,x_k-1)x_k where k >1 is a fixed integer. Then(1) N jorms an ideal of R,(2) R = N + R₁, where R₁ is isomorphic to a subdirect sum of fields. In particular, if N is commutative, then R is commutative.