A module M is said to be lifting if, for any submodule X of M, there exists a decomposition M = A ⊕ B such that A ⊆ X and X/A is a small submodule of M/A. A lifting module is defined as a dual concept of the extending module. A module M is said to have the finite internal exchange property if, for any direct summand X of M and any finite direct sum decomposition M = M1 ⊕⋯ ⊕ Mn, there exists a direct summand Ni of Mi (i = 1,…,n) such that M = X ⊕ N1 ⊕⋯ ⊕ Nn. 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 d-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.

中文翻译：

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具有有限内部交换特性和空心模块直接和的提升模块

一个模块米据说是起重如果，对于任何子模块X的米, 存在分解米 = 一种 ⊕ 乙这样一种 ⊆ X和X/一种是一个小的子模块米/一种. 提升模块被定义为扩展模块的双重概念。一个模块米据说有有限内部交换性质如果，对于任何直接求和X的米和任何有限直接和分解米 = 米1 ⊕⋯ ⊕ 米n, 存在直接加法ñ一世的米一世 (一世 = 1,…,n)这样米 = X ⊕ ñ1 ⊕⋯ ⊕ ñn. 本文关注提升模块的以下两个基本未解决问题：“对所有提升模块都具有有限内部交换性质的环进行分类”和“不可分解提升模块的直接和何时提升？”。在本文中，我们证明任何d- 右完美环上的无平方提升模满足有限内交换性质。此外，我们给出了右完美环上空心模的直接和在有限内交换性质下被提升的充要条件。