An R-module M is called strongly D2 if M(n) is a D2 (equivalently, direct projective) module for every positive integer n. In this paper, we consider the class of quasi-projective R-modules, the class of strongly D2 R-modules and the class of D2R-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li–Chen–Kourki. We also give structural characterizations of strongly D2 modules for finitely generated modules over a principal ideal domain. In addition, we characterize some rings such as Artinian semisimple rings, hereditary rings, semihereditary rings and perfect rings in terms of strongly D2 modules.

中文翻译：

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Strongly D2 模块及其应用

一个R-模块米被强烈称为D2如果米(n)是一个D2每个正整数的（等效地，直接投影）模块n. 在本文中，我们考虑准射影类R-modules，强类D2 R-modules 和类D2R-模块。我们首先表明这些类别是不同的，这对 Li-Chen-Kourki 提出的问题给出了否定的答案。我们还给出了强烈的结构特征D2modules 用于在主要理想域上有限生成的模块。此外，我们对一些环，如阿蒂尼环半单环、遗传环、半遗传环和完美环进行了强烈的表征。D2模块。