The Baer–Kaplansky Theorem for All Abelian Groups and Modules

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-01-05 DOI:10.1142/S1664360721500053

Simion Breaz, Tomasz Brzezi'nski

引用次数: 3

Abstract

It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.

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所有阿贝尔群和模的Baer-Kaplansky定理

证明了Baer-Kaplansky定理可以推广到所有的阿贝尔群，只要群的自同态环被相应堆的自同态桁架所代替。也就是说，每一个阿贝尔群都是由它的自同构桁架确定为同构的，并且与某些阿贝尔群相关联的[公式:见文]和[公式:见文]的两个自同构桁架之间的每一个同构都是由[公式:见文]和[公式:见文]之间的同构以及[公式:见文]中的一个元素引起的。然后通过考虑模块堆将这种对应关系扩展到环上的所有模块。证明了与一个模[公式:见文]相关联的堆的自同态桁架决定了[公式:见文]是其自同态环上的一个模。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.