Algebraic Extensions of Commutative Regular Rings
Algebraic Extensions of Commutative Regular Rings
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交换正则环的代数扩展
DOI：
10.4153/cjm-1970-132-4
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发表时间：
1970
期刊：
影响因子：
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通讯作者：
R. Raphael
中科院分区：
文献类型：
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作者：
R. Raphael
本文研究了交换半素环的代数闭包。然而，主要感兴趣的是冯·诺依曼意义下的正则环。它们对于半质环的作用与场对于整环的作用相同。定义了两个基本不同的概念：“代数”扩张和“弱代数”扩张。每一个都具有传递性，并产生一个闭包，该闭包在同构之前是唯一的，并且是“通用的”。两者在田野中重合。这里称为“代数”的扩展是由Enochs[5]和我独立研究的。我们关于这些推广的结果从一个不同的角度出发，并允许我们回答Enochs提出的一个问题。此外，为了得到弱代数闭包，这些结果是必需的(并且是发展的)，这是所寻求的原始闭包。弱代数扩张的动机可以在Shoda[14，第134页，第1号]的工作中找到。
In this paper we study algebraic closures for commutative semiprime rings. The main interest, however, is with rings which are regular in the sense of von Neumann. These play the same role with respect to semiprime rings as fields do with respect to integral domains. Two generally distinct notions are defined: “algebraic” and “weak-algebraic” extensions. Each has the transitivity property and yields a closure which is unique up to isomorphism and is “universal”. Both coincide in fields. The extensions here called “algebraic” were studied independently by Enochs [5] and myself. Our results on these extensions proceed from a different point of view, and allow us to answer a question posed by Enochs. Furthermore, these results are required (and were developed) in order to obtain the weak-algebraic closure, which was the original closure sought. The motivation for the weak-algebraic extensions is found in the work of Shoda [14, p. 134, no. 1].