Lie isomorphisms of prime rings
Lie isomorphisms of prime rings
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DOI：
10.1090/s0002-9947-1969-0251077-5
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发表时间：
1969-08
影响因子：
1.3
通讯作者：
W. S. Martindale
中科院分区：
文献类型：
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作者：
W. S. Martindale
对于所有的x，y，E，S。我们对环的李同构的研究的兴趣和观点最初（现在仍然）是由I。N. Herstein关于将全矩阵环的李结构的经典定理推广到任意简单环的李结构的结果。在我们的例子中，出发点是认识到应该有可能推广L的下列定理。华：除环（n >2，特征为& 2，3）上所有n × n矩阵组成的环R的每个李自同构都具有a + -r的形式，其中a是R的自同构或反自同构的负数，X是R到其中心的加法映射，该加法映射将自同构映射到零。实际上，利用华的一些技巧和Nathan Jacobson的一些有价值的建议，我们在[2]中，粗略地说，能够在较弱的拓扑下得到相同的结论，即R仅仅是一个具有三个正交幂等元且其和为1的本原环。在最近的一篇论文[3]中，在更强的假设R是单的同时，我们能够将幂等元的数量从3减少到2。在大多数情况下，同样的技术被用于这第二篇论文，虽然张量积方法由于雅各布森是取代繁琐的计算涉及矩阵单位由于华，和一些结果Herstein李理想的简单环似乎是必要的。本文的目标是定理11，将上述结果推广到R是具有两个正交幂等元和为1的素环的情形。素性是单纯性和非单纯性的自然概括，而且，在保持自由的根式和理想的（次）直和的意义上，它也许是人们可以做出的最强的概括。幂等元的假设是否必要仍然是一个主要的悬而未决的问题。在我们所有的工作主题（包括本文件），我们的论点很大程度上依赖于一个非平凡的幂等元的存在。成功地去除幂等元的假设肯定需要全新的方法;例如，人们将不得不面对任意除环的情况。
for all x, y E S. Our interest and viewpoint toward the study of Lie isomorphisms of rings was originally (and still is) inspired by the work done by I. N. Herstein on generalizing classical theorems on the Lie structure of total matrix rings to results on the Lie structure of arbitrary simple rings. In our case the starting point was the realization that it should be possible to extend the following theorem of L. Hua [1]: every Lie automorphism of the ring R of all n x n matrices over a division ring, n >2, characteristic & 2, 3, is of the form a + -r, where a is either an automorphism or the negative of an antiautomorphism of R and X is an additive mapping of R into its center which maps commutators into zero. Indeed, using some of Hua's techniques and some valuable suggestions due to Nathan Jacobson, we were in [2], roughly speaking, able to obtain the same conclusion under the weaker assumptiop that R was merely a primitive ring possessing three orthogonal idempotents whose sum was 1. In a recent paper [3], while making the stronger assumption that R was simple, we were able to lower the number of idempotents from three to two. For the most part, the same techniques were used in this second paper, although a tensor product method due to Jacobson was to replace tedious calculations involving matrix units due to Hua, and some results of Herstein on Lie ideals of simple rings seemed necessary. Our goal in this paper is Theorem 11, in which we extend the above results to the situation where R is a prime ring with two orthogonal idempotents whose sum is 1. Primeness is a natural generalization of simplicity and primitivity, and, in the sense of keeping free of the radical and of (sub)direct sums of ideals, it is perhaps the strongest generalization one may make. Whether the assumption of idempotents is necessary or not is still a major open question. In all our work on the subject (including the present paper) our arguments rest heavily on the presence of a nontrivial idempotent. A successful removal of the assumption of idempotents would certainly require totally new methods; one would, for example, have to face the situation of an arbitrary division ring.