Outer Automorphisms of Semi‐Prime Rings
Outer Automorphisms of Semi‐Prime Rings
复制标题
半素环的外自同构
DOI：
10.1112/jlms/s2-18.2.209
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发表时间：
1978
影响因子：
1.2
通讯作者：
S. Montgomery
中科院分区：
文献类型：
--
作者：
S. Montgomery
最近有大量的工作是关于环R的结构和它的固定环RG在R的自同构的有限群G的作用下的结构之间的关系。这些结果中的许多要求群的阶\G\不作为R上的零因子，因为它们依赖于G的一个基本定理。Bergman和IM Isaacs [3]指出，在此假设下，如果RG是幂零的，则R是幂零的。这类结果的例子（假设R没有加性|G|- 扭转）如下：如果R是半素的，则R是Goldie环当且仅当RG是Goldie环（这是由VK Kharchenko [9]证明的，M. Cohen [4]）;若RG满足多项式恒等式（PI），则R也满足PI（这也是Kharchenko [9]的一个结果）;若R是半素的且RG是Noether的，则R是Noether的（这是D. Farkas和R. Snider [6]）.本文在不设G作用于R的条件下，证明了当R是半素的且G由某些外自同构组成时的类似定理. G将不会简单地由R上的非内部自同构组成，而是满足Kharchenko在他对具有自同构的广义恒等式的研究[10]中给出的外部自同构的定义。他的定义，任何半质环R，使用环的同素R？相对于过滤器！F的双边本质理想的R，当R是素的，它只是意味着G是外部时，扩大到R？。我们称G^-outer为Kharchenko定义的outer。我们首先研究#"-outer与其他outer概念的关系。我们证明了一个群G是^"-外部的在形式上弱于它是”完全外部的”，即Y的定义。Miyashita [13]在非对易伽罗瓦理论中使用。当R是半素Goldie且具有一个经典的同素环Q（R）时，我们证明了G在R上是jF-外的当且仅当当推广到Q（R）时，G是完全外的.特别地，当R是素Goldie时，G将是Q（R）上的通常意义上的外部。我们还更明确地证明了当R是具有极小单侧理想的本原环时，或者更一般地当R是中心闭包具有极小单侧理想的素环时，G是^"-外环意味着什么。
There has been a great deal of work recently concerning the relationship between the structure of a ring R and the structure of its fixed ring RG under the action of a finite group G of automorphisms ofR. Many of these results require that the order\G\of the group not act as a zero-divisor on R, as they depend on a fundamental theorem of G. Bergman and IM Isaacs [3] which states that, under this hypothesis, if RG is nilpotent then R is nilpotent. Examples of such results (assuming that R has no additive| G|-torsion) are as follows: if R is semi-prime, then R is a Goldie ring if and only if RG is a Goldie ring (this was proved by VK Kharchenko [9] and independently for solvable groups by M. Cohen [4]); if RG satisfies a polynomial identity (PI), then R also satisfies a PI (this is also a result of Kharchenko [9]); and if R is semi-prime and RG is Noetherian, then R is Noetherian (this is a theorem of D. Farkas and R. Snider [6]).In this paper, we will prove analogous theorems, with no assumptions about\G\acting on R, in the situation when R is semi-prime and G consists of certain outer automorphisms. G will not simply consist of automorphisms which are not inner on R, but instead will satisfy the definition of outer given by Kharchenko in his study of generalized identities with automorphisms [10]. His definition, for any semi-prime ring R, uses the ring of quotients R? of it relative to the filter! F of two-sided essential ideals of R, and when R is prime it simply means that G is outer when extended to R?. We will call G^-outer if it is outer by Kharchenko's definition. We first examine the relationship of#"-outer to other notions of outer. We show that a group G being^"-outer is formally weaker than its being" completely outer", the definition of Y. Miyashita [13] used in non-commutative Galois theory. When R is semi-prime Goldie with a classical ring of quotients Q (R), we show that G is jF-outer on R if and only if, when extended to Q (R), G is completely outer. In particular, when R is prime Goldie, G will be outer in the usual sense on Q (R). We also show more explicitly what it means for G to be^"-outer when R is primitive with a minimal one-sided ideal, or more generally when R is a prime ring whose central closure has a minimal one-sided ideal.