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Cotorsion pair extensions. (English) Zbl 1215.16016

Summary: Assume that \(S\) is an almost excellent extension of \(R\). Using functors \(\operatorname{Hom}_R(S,-)\) and \(-\otimes_RS\), we establish some connections between classes of modules \(\mathcal C_R\) and \(\mathcal C_S\), cotorsion pairs \((\mathcal A_R,\mathcal B_R)\) and \((\mathcal A_S,\mathcal B_S)\). If \(\mathcal C_S\) is a \(T\)-extension or (and) \(H\)-extension of \(\mathcal C_R\), we show that \(\mathcal C_S\) is a (resp., monomorphic, epimorphic, special) preenveloping class if and only if so is \(\mathcal C_R\). If \((\mathcal A_S,\mathcal B_S)\) is a \(TH\)-extension of \((\mathcal A_R,\mathcal B_R)\), we obtain that \((\mathcal A_S,\mathcal B_S)\) is complete (resp., of finite type, of cofinite type, hereditary, perfect, \(n\)-tilting) if and only if so is \((\mathcal A_R,\mathcal B_R)\).

MSC:

16S20 Centralizing and normalizing extensions
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16D90 Module categories in associative algebras
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