Abstract
這篇論文中,[u,v]=uv-vu,而且R是一個含有對合函數的質環,此外K,S分別表示R中所有反對稱和對稱元素的集合,A是R中的一個right ideal.我們將考慮在R上的微分算子d及由K到K的加性函數的一些性質.Theorem A. 如果d是R的微分算子,滿足[[d(x^n1),x^n2],...,x^nt]=0 對於所有K(或S)中的元素. 則只有兩種情形會發生,不是d=0就是R滿足4個變數的標準多項式.Theorem B. 如果f是由K到K的加性函數,滿足[f(x),x^m]=0 對於所有的x in K.則[f(x),x]=0對於所有的x in K.條件是R的特徵值等於0或大於m而且R不是一個4維單純代數的order.Theorem C. 如果f是一個由非零right ideal A 到R的加性函數,滿足f(x)^2=x^2對於所有x in rightideal A. 則f是right ideal A 上的一個identity函數或負identity函數.條件是R的特徵值不等於2而且right ideal A 的lie product乘上right ideal A不等於0(that is [A,A]A is not zero).Theorem D. 是ThmB and ThmC 的一個應用.關於本篇的內容第一章 簡介第二章 預備知識第三章 廣義反對稱元超中心第四章 反對稱元上的加性函數第五章 函數方程In this article we denote [u,v]=uv-vu. Theorem A. Let R be aprime ring with involution and d a derivation of R such that[[d(x^n1),x^n2],...,x^nt]=0 for all skew elements x in K (respS) thend=0 or R satisfies standard polynomial in 4 variables,where n1,...,nt are fixed natural numbers.Theorem B. Let R be aprime ring with involution and f is an additive map fromK into Ksatisfying [f(x),x^m]=0 for all x in K, where m is a fixednatural number.Then unless R is an order in a 4-dimensionalcentral simple algebra, [f(x),x]=0 for all x in K, provided thatcharR=0 or >m.Theorem C. Let R be a prime ring with the nonzeroright ideal A, extended centroidC and f is an additive map fromA into R. Suppose that charR is not 2 and [A,A]Ais not zero. Iff(x)^2=x^2 for all x in A then either f=I or f=-I, where I isthe identity on A.Theorem D. Let R be a prime ring withinvolution and charR is not 2. Suppose that f is an additive mapfrom K to K such that f(x)^2=x^2 for all x in K.then f=I or f=-Ion K, unless R is an order in a 4-dimensional central simplealgebra.