Lifting modules with finite internal exchange property and direct sum of hollow modules
Lifting modules with finite internal exchange property and direct sum of hollow modules
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具有有限内部交换性质的提升模块和空心模块的直和
DOI：
10.1142/s0219498821501899
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发表时间：
2021
影响因子：
0.8
通讯作者：
Yosuke Kuratomi
中科院分区：
文献类型：
--
作者：
Isao Kikumasa;Yosuke Kuratomi;Yoshiharu Shibata;Yosuke Kuratomi
一个模称为Belifting，如果对于的任何子模，存在一个分解使得和是的一个小子模。提升模被定义为扩张模的对偶概念。一个模具有有限内部交换性质，如果对任意的直和和分解，存在一个直和使得.本文讨论了提升模的两个基本未解决的问题：“对提升模都具有有限内部交换性质的环进行分类”和“不可分解提升模的直和何时是提升的？".本文证明了右完全环上的任意无平方提升模满足有限内交换性质。另外，我们给出了右完全环上的空模的直和具有有限内交换性质的提升的充要条件。
A moduleis said to beliftingif, for any submoduleof, there exists a decompositionsuch thatandis a small submodule of. A lifting module is defined as a dual concept of the extending module. A moduleis said to have thefinite internal exchange propertyif, for any direct summandofand any finite direct sum decomposition, there exists a direct summandofsuch that.This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.