Right noetherian rings integral over their centers
Right noetherian rings integral over their centers
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DOI：
10.1016/0021-8693(73)90173-7
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发表时间：
1973-10
影响因子：
0.9
通讯作者：
W. D. Blair
中科院分区：
文献类型：
--
作者：
W. D. Blair
右诺特环是在其中心的子环上积分的环，它具有交换诺特环的许多性质。我们开始在第一节回顾的定义和基本性质的非交换环，这是积分的中心。在这里我们还陈述了一个著名的小定理，这是必不可少的续集。第2节包含的主要定理的文件，即一个右诺特环积分超过其中心是嵌入在一个artinian环，有交集的权力，其Jacobson根等于零，并有下降链条件素理想。当这类商环存在时，研究了这类环的经典商环的性质。第三部分包括一些可能发生的病理学的例子。符号和惯例。假设所有的环都有单位元。对于x EI？，Y（X）=（aER [xa= 0）称为X的右零化子。类似地，x的左零化子是Z（x）=（GEW j ax= 01。如果I（x）= T（X）= 0，则称x是正则的。设Q（A）d表示A的右经典商环，J（A）表示A的Jacobson根，Z（A）表示A的中心.最后，* 表示正整数。
Right noetherian rings which are integral over subrings of their centers enjoy many of the properties of commutative noetherian rings. We begin in Section I by recalling the definitions and elementary properties of noncommutative rings which are integral over their centers. Here we also state a well known theorem of Small which is indispensable in the sequel. Section 2 contains the main theorems of the paper; namely that a right noetherian ring integral over its center is embeddable in an artinian ring, has the intersection of the powers of its Jacobson radical equal to zero, and has the descending chain condition on prime ideals. Properties of classical quotient rings of such rings are also studied, when such quotient rings exist. Section 3 consists of examples of some of the pathoIogy which may occur.Notations and Conventions. All rings are assumed to have unit element. For x EI?, Y (X)=(a ER [xa= 0) is called the right annihilator of X. Similarly the left annihilator of x is Z (x)=(GEW j ax= 01. If I (x)= T (X)= 0 then x is said to be regular. We let Q (A) d enote the right classical quotient ring of A; J (A) the Jacobson radical of A and Z (A) the center of A. Finally* denotes the positive integers.