On the Lie and Jordan Rings of a Simple Associative Ring
On the Lie and Jordan Rings of a Simple Associative Ring
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DOI：
10.2307/2372531
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发表时间：
1955-04
影响因子：
1.7
通讯作者：
I. Herstein
中科院分区：
文献类型：
--
作者：
I. Herstein
给定任意结合环A，我们可以用它的运算和元素构成两个新的环。它们使用A的元素和A中定义的加法，但引入了新的乘法来使它们成为环，尽管不一定是结合环。第一个是A的李环AL，它使用一个乘法定义为：[a，B] =ab ba，对于任何a，B e A，其中ab是A中元素的普通结合积。其中第二个是A的Jordan环A '，它的乘法定义为aoba B + ba，对于A中的任意一对元素a，B。由于定义的方式决定性地依赖于A的结合积，因此很自然地可以预期，在这两个新环的结构与A的结构之间应该存在着密切的关系。本文研究了这种关系的一个阶段，即结合环A的理想结构与Lie环和Jordan环AL和A'的理想结构之间的联系。更具体地说，我们研究如何简单的结合环A反映到类似的性质AL和A，。当我们说U是Al的理想时，或者等价地说，当我们说U是A的Jordan理想时，我们的意思是U是A的加法子群，并且对任何xeU和任何是的，xoy=xy+yx是U的元素。我们类似地定义了A的李理想和AL的理想。虽然本文的主要结果处理的情况下，A是一个简单的环，许多其他的结果并不需要简单的假设，以保持有效;所以，除非另有说明，我们不作简单的假设为A。
Given any associative ring A we can form, using its operations and its elements, two new rings. These use the elements of A and the addition as defined in A, but new multiplications are introduced to render them rings, albeit not necessarily associative rings. The first of these, the Lie ring AL of A uses a multiplication defined by [a, b] =ab ba for any a, b e A where ab is the ordinary associative product of elements in A. The second of these, the Jordan ring of A, A', has its multiplication defined by ao bab + ba for any pair of elements a, b in A. Being defined in a manner so decidedly dependent on the associative product of A, it is natural to expect that an intimate relationship should exist between the structure of these two new rings and that of A. In this paper we study one phase of this relationship, namely the connection between the ideal structure of A as an associative ring with the ideal structure of AL and A' as Lie and Jordan rings respectively. To be more specific, we investigate how simplicity of A as an associative ring reflects into analogous properties of AL and A,. When we say that U is an ideal of Al, or, equivalently, when we say that U is a Jordan ideal of A, we mean that U is an additive subgroup of A and that for any xeU and any yeA, xoy=xy+yx is an element of U. We similarly define Lie ideals of A and ideals of AL. Although the main results of this paper deal with the case in which A is a simple ring, many of the other results do not require the assumption of simplicity in order to remain valid; so, unless otherwise stated, we make no assumption of simplicity for A.