Bijective Extensions of Injective Ring Endomorphisms
Bijective Extensions of Injective Ring Endomorphisms
复制标题
DOI：
10.1112/jlms/s2-25.3.435
复制
发表时间：
1982-06
影响因子：
1.2
通讯作者：
D. Jordan
中科院分区：
文献类型：
--
作者：
D. Jordan
设R是结合环，a：R->；R是环单态。我们可以构造一个溢环A=A(R，a)，它在某种意义上是a作为自同构扩张到的R的极小溢环。本文的目的是研究环R和A之间的关系，特别注意链条件，并着眼于在斜多项式环理论中的应用。有简单的例子表明，环A，R中的任何一个是Notherian的，而另一个是Notherian的。这些例子作为例子3.4、5.8和5.9出现在论文中。然而，我们将证明，如果R是具有单位元的左Artin，则A是左Artin的。在本文的过程中，我们将找到A是左Notherian(分别是左Artin)的充要条件。
Let R be an associative ring and a: R-> R be a ring monomorphism. One can construct an overring A= A (R, a) which is, in a sense, the minimal overring of R to which a extends as an automorphism. It is the purpose of this paper to study the relationship between the rings R and A with particular attention being paid to chain conditions and with a view to applications in the theory of skew polynomial rings.Easy examples exist which show that it is possible to have either of the rings A, R left Noetherian without the other being left Noetherian. Such examples appear in the paper as Examples 3.4, 5.8 and 5.9. However we shall show that if R is left Artinian with identity then A is left Artinian. In the course of the paper we shall find necessary and sufficient conditions for A to be left Noetherian (respectively left Artinian).