Γ－环的强诣零根研究

RESEARCH ON THE STRONGLY NIL RADICAL OF Γ-RINGS

报道作者在Γ－环的强诣零根方面的综合研究。Γ－环有无强诣零根的问题，来自于１９７１年Ｃｏｐｐａｇｅ与Ｌｕｈ的“Ｒａｄｉｃａｌｓｏｆｇａｍｍａｒｒｉｎｇｓ”中的定理５．４，由此可得：任何Γ－环都有强诣零根，这是一个严重的误解。但定理５．４的证明存在问题，因此就开始了Γ－环有无强诣零根的专项研究。先用性质接近于强诣零根的、且一定存在的拟次强诣零根、拟强诣零根去取代存在与否尚属未知的强诣零根，在研究Γ－环的结构上，它们起到了类似于强诣零根的作用；接着在给Γ－环略增条件后称之为Ｗｅａｋｅｒ Γ＿Ｎ环中，证明了一定存在强诣零根，在这个环中还对强诣零根进行了多种刻划；以后在有限个元素的特殊Γ－环中进行了研究，应用动力系统中关于有限型子转移的若干结果，证得结论：有限个元素的Γ－环一定存在强诣零根；最后，成功地构造出一个反例一不存在强诣零根Γ－环．从而，否定了Ｃｏｐｐａｇｅ与Ｌｕｈ的定理５．４，澄清了Γ－环的根论研究史上的一个严重误解。

t was a summary report of the author's work on Strongly Nil Radical of Γ-Ring was given. The problem whether the Strongly Nil Radical ofΓ-Ring always exists,deriyed from the reference 'Radicals of gamma rings' by Coppage and Luh in 1971. From the Theorem 5.4 of that paper, one can get the conclusion readily: There must exist a strongly nil radical in an arbitrary Γ-Ring M. However, there was some essential mistake in the proof of Theorem 5.4. From then on, the specific program on Existence of Strongly Nil Radical was carried out. At first, some radicals so called the Quasi-Strongly Nil Radical and Quasi-Substrongly Nil Radicalwith the property close to that of Strongly Nil Radical were used to take place of theStrongly Nil Radical whose existence was still undecided. Those replacement played thesimilar role as that of Strongly Nil Radical, for the research on structure of Γ-Rings.And then, the positive answers were given to the problem, in two cases that if M isa weaker Γ_N-ring or a Γ-Ring of finite elements, then there must exist a strongly nilradical in M. For the first case, various descriptions were obtained for Strongly Nil Radical of weaker Γ_N-Ring,a new subclass of Γ- Ring that it was defined. In theproof of second case, shift of finite type was made use of in Dynamical System.Finally, an elaborate counterexample was successfully consmicted, that is,a Γ-Ring without Strongly Nil Radical, and thus the Th.5.4 of Coppage and Luh was denied.Therefore the serious misunderstanding in the history for radical research of Γ-rings was clarified.