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The characterizations on \(\mathcal{P}_n\)-injective modules. (Chinese. English summary) Zbl 1363.16009

Summary: Let \(R\) be a ring, and \(n\) a fixed nonnegative integer. An \(R\)-module \(D\) is called a \(\mathcal{P}_n\)-injective module if Ext\(_R^1(P, D)=0\) or any \(R\)-module \(P\) has projective dimension at most \(n\). In this paper, we prove that, \((\mathcal{P}_n, \mathcal{D}_n)\) is a hereditary cotorsion theory, where \(\mathcal{P}_n\) is the class of all \(R\)-modules with projective dimension at most \(n\), and \(\mathcal{D}_n\) is the class of all \(\mathcal{P}_n\)-injective modules. It is also shown hat every \(\mathcal{P}_n\)-injective module is injective if and only if gl. dim\((R)\leq n\). Finally, for \(n\geq1\), we prove that every \(R\)-module is \(\mathcal{P}_n\)-injective if and only if 1. FPD\((R)=0\).

MSC:

16D50 Injective modules, self-injective associative rings
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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