Absolutely pure modules
Absolutely pure modules
复制标题
DOI：
10.1090/s0002-9939-1967-0224649-5
复制
发表时间：
1967
期刊：
影响因子：
--
通讯作者：
B. H. Maddox
中科院分区：
文献类型：
--
作者：
B. H. Maddox
导论.本文给出了绝对纯模的几个性质。证明了Dedekind环上模的绝对纯性与内射性是等价的。然而，它被证明是绝对纯和内射不是等价的性质模环上的非诺特。证明了每个模都有极大绝对纯子模。给出了极大绝对纯子模唯一的一个充分条件。本文是作者在南卡罗来纳州大学获得合作研究生奖学金的博士论文的一部分。作者感谢埃德加伊诺克教授提出这个话题，并指导其发展，同时提供了足够的灵感和帮助，最重要的是，表现出无限的耐心。在本文中，所有环都有一个单位，所有模都是么正的。A总是表示一个环。我们同意，如果E'是E的子模，并且v. E'->E是正则注入，则映射1 ®v：F®E' ->P E将被称为正则映射，其中1：F-^F是P的恒等映射。如果正则映射是对所有P的注入，则E'被称为是P的纯子模。观察到如果E'是E的纯子模，则aEC\E' =aE'对于所有非零aEA，通过检查图
Introduction. In this paper several properties of absolutely pure modules are given. It is shown that absolutely pure and injective are equivalent properties for modules over Dedekind rings. However, it is proved that absolutely pure and injective are not equivalent properties for modules over rings which are not Noetherian. That every module has a maximal absolutely pure submodule is also established. A sufficient condition for the uniqueness of a maximal absolutely pure submodule is also given. This paper constitutes a portion of the author's doctoral dissertation written at the University of South Carolina where he held a Cooperative Graduate Fellowship. The writer is indebted to Professor Edgar Enochs who suggested this topic and directed its development while providing sufficient inspiration and assistance and, most of all, exhibiting infinite patience. In this paper all rings will have a unit and all modules will be unitary. A will always denote a ring. We agree that if E' is a submodule of E and v. E'—>E is the canonical injection then the map 1 ®v: F®E' —>P £ will be called the canonical map where 1: F-^F is the identity map of P. If the canonical map is an injection for all P, then E' is said to be a pure submodule of P. Observe that if E' is a pure submodule of E then aEC\E' =aE' for all nonzero aEA by examining the diagram