Covers, preenvelopes, and purity
Covers, preenvelopes, and purity
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发表时间：
2006-11
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通讯作者：
Henrik Holm;Peter Jørgensen
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作者：
Henrik Holm;Peter Jørgensen
证明了如果一类模在纯商下是闭的，那么它是预覆盖的当且仅当它是覆盖的，当且仅当它是直和闭的。这是受Rada和Saor在文[1]中的一个对偶结果的启发.我们还证明了：如果一类模包含地环，且在扩张、直和、纯子模和纯商下是闭的，则它构成Salce提出的所谓完全余扭对的前半部分；这比覆盖更强.给出了一些具体的模类的应用，如扭对中的同调函子核和无挠模.
We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saor\'in. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it forms the first half of a so-called perfect cotorsion pair as introduced by Salce; this is stronger than being covering. Some applications are given to concrete classes of modules such as kernels of homological functors and torsion free modules in a torsion pair.